Preheat the oven to 400°F. Place a large oven-proof skillet or sauté pan over medium heat and add the olive oil and butter; heat until the butter is melted and the fat is shimmering.

Add the garlic cloves to the pan and cook, turning occasionally, until lightly browned, about 3–5 minutes. Transfer the browned garlic to a plate, leaving the oil and butter in the pan.

Pat the chicken thighs very dry with paper towels. Season both sides of the thighs with salt and pepper.

Increase the burner to medium-high. If you have a splatter screen, use it. Place the chicken thighs in the skillet skin-side down and cook without moving until the skin is browned and crisp, about 5–6 minutes. Remove the skillet from the heat.

Strip the leaves from the rosemary and thyme stems and sprinkle the fresh herbs over both sides of the chicken. Flip the thighs so the skin side is facing up, and scatter the reserved browned garlic cloves around the chicken.

Transfer the skillet to the preheated oven and roast until the skin is crispy and the internal temperature of the chicken reaches 165°F, about 20 minutes.

While the chicken roasts, whisk the chicken broth and cornstarch together in a small bowl until smooth.

When the chicken is done, remove the skillet from the oven. Transfer the chicken and garlic to a plate and keep warm. Place the skillet with the pan drippings over medium heat and bring to a simmer.

Whisk the broth–cornstarch mixture again, then pour it into the skillet. Cook, whisking constantly, until the sauce thickens, about 1 minute. Remove from the heat and stir in lemon juice to taste.

Spoon some pan sauce onto plates, top with a roasted chicken thigh and several of the roasted garlic cloves, and finish by sprinkling with fresh parsley.