Make the brine: Bring 1 1/2 gallons water and 1/3 cup salt to a boil in a large pot, whisking until the salt dissolves. Remove from the heat. Add the lemon halves, tarragon, parsley, garlic, bay leaves and peppercorns; let cool completely.
Place the hens breast-side down on a cutting board. Use kitchen shears to cut along either side of the backbone to remove. Turn the hens breast-side up and open so the drumsticks point out; press the breastbone firmly to flatten. Submerge the hens in the brine and refrigerate at least 6 hours and up to 12 hours.
Remove the hens from the brine and rinse well; pat dry. Place breast-side up in a baking dish and refrigerate, uncovered, to dry out the skin, at least 30 minutes.
Sprinkle the hens with salt. Wrap 2 bricks with foil. Heat 1 tablespoon each olive oil and butter in each of 2 large nonstick or cast-iron skillets over medium-high heat. Add 1 hen, skin-side down, to each skillet and scatter the thyme and garlic around it. Top each hen with a foil-wrapped brick and sear until beginning to brown, about 20 minutes, rotating the pans as needed for even cooking. Remove the bricks and carefully flip the hens (do not replace the bricks); continue cooking until a thermometer inserted into the thickest part of the thigh registers 165 degrees F, 20 to 25 more minutes. Cut each hen in half and serve with the romesco and chimichurri sauces.
Toss 1/2 pound plum tomatoes and 4 garlic cloves (unpeeled) with 2 tablespoons olive oil and 1/4 teaspoon kosher salt; spread on a baking sheet. Roast at 350 degrees F until soft, 20 minutes; let cool, then squeeze out the garlic. Combine the tomatoes, garlic, 3/4 cup chopped roasted red peppers, 1/4 cup sherry vinegar, 1 tablespoon honey, 1/2 teaspoon kosher salt and a pinch of red pepper flakes in a blender; blend until smooth. Transfer to a bowl; stir in 1/2 cup chopped toasted almonds. Slowly drizzle in 1/3 cup olive oil, whisking.
Put 6 garlic cloves (unpeeled) on a sheet of foil; drizzle with olive oil and seal into a packet. Roast at 350 degrees F until soft, about 20 minutes; let cool. Squeeze out the garlic cloves and mash into a paste. Combine 1 bunch basil (cut into thin ribbons), 1 bunch finely chopped parsley, 1/4 bunch finely chopped oregano, 1/3 cup sherry vinegar, 2 finely chopped shallots, 1/2 teaspoon kosher salt, and pepper to taste. Slowly drizzle in 1/3 cup olive oil, whisking.