Put chicken, garlic, bay leaf, peppercorns, and a few parsley sprigs in a Dutch oven. Fill almost to the top with cold water; bring to a boil. Reduce to a bare simmer. Cook until chicken is no longer pink in the center, 18 to 20 minutes; transfer to a bowl. Add enough poaching liquid to cover; let cool. Reserve garlic and 1/2 cup remaining poaching liquid from pot.
Process tarragon, bread, capers, reserved poached garlic, and parsley in a food processor until smooth, about 15 seconds. With processor running, add oil and reserved 1/2 cup poaching liquid. Stir in salt, and season with pepper.
Bring a small pot of water to a boil; add peas. Cook fresh peas until bright green and crisp-tender (or frozen peas until heated through), about 2 minutes; drain. Spoon 2 tablespoons sauce onto each of four plates. Cut each breast half into 5 or 6 pieces. Divide among plates; scatter peas on top. Garnish with tarragon; serve with extra sauce on the side.