You can cook peeled and unpeeled shrimp using the same methods, but the shell prevents the flesh from browning. For example, if you like the smoky flavor of the grill but don't like the char it leaves, grill the shrimp with the shells on and peel them after grilling. Or, if you want to cook the shrimp quickly over high heat but don't want the browned color and texture a sear imparts, saute the shrimp unpeeled and pull the shell off later. You can add uncooked shells to fish stock if you use them the same day you peeled them.
Heat a couple tablespoons of oil in a saute pan over medium heat. Pat the shrimp with a paper towel to dry them.
Add pungents and aromatics, such as minced garlic and shallots, to the pan and cook until softnened, if desired. You can also add whole herb sprigs and whole spices to flavor the oil if you like.
Add the shrimp to the pan in an even layer and season it with kosher salt and freshly ground black pepper. Cook the shrimp for about 1 minute.
Add about 1/2 cup of dry white wine, water or stock to the pan. Cook the shrimp, turning occasionally, until the liquid evaporates, about 4 minutes.
Pull the shells off the shrimp if not already and discard. Toss the shrimp in the pan to coat with the olive oil and serve.
Add enough stock or water to a saucepan or pot to cover the shrimp by a couple inches. If you want to make a quick court bullion, add aromatic and pungent ingredients, such as mirepoix, lemon slices and garlic, to the water and simmer for 15 minutes.
Season the liquid to taste and bring it to a boil. Add the shrimp to the pan and take it off the heat.
Cover the pan with a tight fitting lid. Cold-poach the shrimp for 15 minutes then drain in a colander. Don't rinse the shrimp, but pull the shells off when they're cool enough to handle and serve with melted butter.
Marinate the shrimp in olive oil and lemon juice for 15 minutes before grilling them, if desired. Skewer the shrimp crosswise with wooden skewers, spacing each about 1/2 inch apart from the next.
Arrange the items on the grill so it has a few inches of space on the left or right side. The edges of the grill aren't as hot as the center, making it the best place for shrimp.
If you're setting up the grill just for shrimp, set it up so it cooks with medium heat, or one layer of charcoal in the charcoal tray.
Wipe off the excess marinade and season the shrimp to taste on both sides if needed. Lay the shrimp on the outer edge of grill if you have other foods on it; if you set the grill to medium, lay the shrimp anywhere on the grill.
Grill the shrimp about 4 minutes on each side. Pull the skewers out and peel the shells before serving.
Peeled shrimp and frozen shrimp have their veins removed unless labelled otherwise. If you have fresh unpeeled shrimp, cut the shell in half down the back then pick out the vein with the tip of a paring knife.