Season chicken with 1 1/4 teaspoons salt and 3/4 teaspoon pepper. Melt 1 tablespoon butter in a large skillet over medium-high heat. Add chicken and cook until deep golden brown, 4 to 6 minutes. Turn, reduce heat to medium, and cook until the internal temperature reaches 165 degrees F, 6 to 8 minutes. Remove to a plate. Increase heat to medium-high. Add shallot to skillet and cook until softened, about 1 minute. Stir in wine and cook, scraping brown bits from bottom of skillet, until reduced by three-quarters, 1 to 2 minutes. Stir in both peas and asparagus and sauté until just tender, 2 to 4 minutes. Stir in garlic and cook until fragrant, about 1 minute. Fold in tarragon and remaining 2 tablespoons butter. Season with salt and pepper. Serve sprinkled with lemon zest.