How best to cook shrimp to use chilled in a composed salad?

Good advice here, starting with the basic "buy good shrimp." The Serious Eats trick with the salt and baking soda brine is interesting too, but I've never felt any need for it. For me, the important part is not over cooking. I watch like a hawk, and take them out of the water, off the grill, or out of the pan the instant they go from translucent to a solid pinkish color.

I also steam them. My first MIL taught me to rinse them, shells on, and then place them in a pot with just the water clinging to them from the rinse. Cover and cook just until they turn pink. That's it! Of course, a few peppercorns and some bay leaves or other seasonings mixed in wont hurt.

I steam them. Shell on or off--it doesn't seem to matter.

Thanks for the tips. I'm gonna try the tips and report back. For what it's worth I buy something called wild American shrimp. I assume it's from Florida. Don't have access to fresh shrimp sadly

p.s. YOU did not cause mushiness in shrimp; it's like they either are or are not mushy due to various factors, but not due to you. Don't buy SEAsian shrimp.

i usually differ greatly from pierino; same thing here. fwiw i always buy shell-on 26-30 count frozen shrimp from Mexico, not Maine (not enough flavor) or Indonesia (mucky). No fancy seasoning needed and some of those can really over-power seafood's delicate favor, incl. shimp, imo(Old Bay be damned). In a small pot of 1-2 qts.water, place shrimp shells(i hope you started w/ shell-on shrimp), some peppercorns , allspice berries, chunks of onion and lemon, 1-2 bay leaves. simmer 10 -30 minutes. remove solids; add defrosted shell-off shrimp, stir, simmer a few minutes til shrimp become opaque. remove shrimp from water and place in bowl of ice water. stir well to cool; remove from water.chill in frig.reduce poaching water to a volume easily stored in your freezer; cool; freeze. Defrost and use for next batch of poached shrimp or as seafood stock for soups and chowders.

Start with a brine of salt and baking soda. It improves the texture of shrimp. SeriousEats found this trick to greatly improve shrimp no matter how you cook it; http://www.seriouseats.com/2015/10/how-to-cook-shrimp-grill-poach-stir-fry-saute.html

You could cook them shell on (heads on too if possible) in a shrimp or crab boil such as Old Bay or Zatarains. Just don't cook them at a rolling boil. Peel them before adding to the salad. Alternatively you could grill them shell on over a wood charcoal fire---not briquettes! But frozen shrimp are definitely part of your problem. Also they probably come from Thailand where they are harvested by slave labor. Always buy fresh if you can, or substitute another crustacean.