Preheat the oven to 425 degrees F. Mix the butter, tarragon, chives, lemon zest and 1/4 teaspoon salt in a small bowl; set aside.
Season the chicken with salt and pepper. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the chicken and cook, turning once, until golden brown, about 5 minutes. Transfer to a baking sheet and bake until cooked through, about 8 minutes.
Meanwhile, wipe out the skillet and add the remaining 1 tablespoon olive oil over high heat. Add the mushrooms and 1/2 teaspoon salt. Cook, stirring occasionally, until golden brown, about 4 minutes. Add the snap peas and shallot; cook, stirring, until the peas are crisp-tender, about 4 minutes. Season with salt and pepper. Top the chicken with the herb butter and serve with the vegetables.