Pearls of wisdom from a performance shop and fellow Lincoln LS owner.
Step1: Take stocker Lincoln LS piston and connecting rod to a place like Oliver racing and have them prototype a set of billet connecting rods. Take the piston to a place like Diamond, or CP or so on and have them make custom pistons for you. Yes it will be expensive but still cheaper (and stronger) than any options mentioned above.
Step2: Do not worry about the stocker crank as it takes allot of power (or detonation) to break a crank, even a cast one.
Step3: Have a knowledgeable head porter port your heads and work on the combustion chamber to open it up a little bit. When done correctly this will drop compression slightly (more boost friendly) which will lead to more safety and also free up some power through better flow. CR will still be high, but not 10.55:1 or 10.75:1 high! CR will still be high enough to get vehicle off the line while not in boost yet and still hit a decent 60' time at the track.
Step4: Call up Level Ten and have them build your stocker 5R55N or 5R55S trannny (depending on which generation of LS) to hold 700+ horsepower. Not that you will make THAT much, but you will have a tranny capable of handling it.
Step5: Get an STS Universal Stage 1 Turbo kit with a T04E turbo if you want to make up to about 375-400 rwhp or the 60-1 turbo if you want to make up to 500rwhp. Retails for $1,995 + shipping. If you are a competent welder and are somewhat mechanically inclined then a remote turbo system is not hard to do. Mike Pietras did well but he messed up in a few areas, namely being the turbo turbine A/R sizing and the oiling system return line size and jetting. Also FULL SCT tuning is available to tune for this turbo system.
Step6: Swap in a Ford GT (supercar) fuel pump if you plan to go to 400rwhp and a KB BAP (Kenne Bell Boost a Pump) if you plan to hit 500rwhp or close. A qualified SCT tuner will have no problem extrapolating voltages from the Ford GT returnless fuel pump voltage table and plugging them into the LS tables to get it close right off the bat. After that the KAM (keep alive memory) will take care of the finetuning the voltages from there. I suggest doing a wiring upgrade between the new GT pump and the FPDM (REM for LS) so as to keep things from overloading. This will also increase the range in which your fuel pump will operate. I usually see a 10-15% drop in fuel pump duty cycle when upgrading wiring in a returnless car. This means an extra 10-15% range for the fuel pump to use before it is maxxed.
Step7: Swap to 03/04 Cobra 39 lb/hr injectors if planning to make 400rwhp or less. If planning to make 500rwhp then swap to Seimens Deka 55lb/hr injectors. You will need the injector adapter harnesses to go from your stocker EV6 connectors to the EV1 style electrical connectors found on the SD 55 lb squirters; the 39's will be drop in the for Gen2 LS. As said before the gen 1 LS's will have to swap to different fuel rails to eliminate the air entrainment feature found in the stocker rails.
Step8: Take your stocker LS slot-in MAFS and fabricate a 3" or 3.5" intake tube to house this in. The stocker LS meter will go up to 400rwhp in a 3" + sized housing and still retain good resolution across the range. If you plan to hit bigger power than 400rwhp then step up to the PMAS HPX slot-in meter.
Step9: Go to Ebay and find a 28"x8"3.5" FMIC and fit it into your front bumper area. It will be a VERY tight fit but you want the most IC that you can get to keep things cool. Some small things will have to be modified but it WILL work.
Step10: Have a good SCT tuner whip up a custom tune for your car. Make sure that they either street tune your car or at a mimimum put it on a loadable dyno (Mustang Dyno or Dynojet w/ Eddie current load cell) and tune yous LS. Unlike superchargers which makes boost more based upon rpm, a turbo makes its boost more directly dependant on load that is placed on the vehicle (engine having to tow weight of vehicle thus creating a load for the engine to pull). In a street tune or on a loaded dyno the engine will see that load and the turbo will spool in a manner similar to real street conditions. If load is correct then things such as VE, spark timing and such will be correct which is VERY critical.
Step11: Go with a custom rear LSD. Others have already started to tackle this but a real result may be coming soon.
I may have missed a small thing or two here or there but that pretty much covers it if you want to build a higher HP LS without having to swap to a different engine. You can also eliminate allot of those steps I stated above if you want to only run about 5-6 psi and make around 350rwhp. Allot of your stocker components will last to that power level as long as you treat your car with some respect. What I just stated above was more geared towards going more extreme than that due to the nature of this particular thread.
I own my own performance shop as well as a highly modded Lincoln LS and have allot of experience with the Lincoln LS and modding it for high performance as well as other vehicles. I am an SCT custom tuner/dealer as well as a dealer for many other places such as K&N, STS turbo, MAC, SLP, and so on and on. That is not meant as any sort of plug for my business, but it does serve to hopefully give what I say credibility as I am not a typical noob in here.

I only prefer to lurk here but I have seen this question, and similar ones to it, asked WAY too many a times without any real definitive answers. Hopefully what I just posted will give some of you a more clearly defined direction in which to go.