Season chicken on both sides with 1 tablespoon salt and 1/2 teaspoon pepper. Preheat a Dutch oven or other large heavy pot over medium-high heat. Add 2 tablespoons butter and the oil to pot. When butter melts and foam subsides, add half the chicken, skin side down, in a single layer; do not crowd pot. (If butter begins to blacken, lower heat.) Fry chicken, turning once, until golden brown on both sides, about 10 minutes total, and transfer to a plate. Repeat with remaining chicken.
Place chicken, skin side up, in a single layer on vegetables; pour juices that have accumulated on plate into pot. Tie parsley sprigs, thyme sprigs, and bay leaf together with kitchen twine; add to pot. Bring to a boil, then reduce to a simmer. Cover partially. Cook until internal temperature of thickest part of chicken registers 165 degrees, 25 to 30 minutes. Transfer chicken to a clean plate. Simmer liquid, uncovered, until reduced slightly, about 5 minutes. Discard herbs.