Preheat the oven to 400°F (205°C).
Heat a combination of olive oil and butter in a large oven-safe skillet over medium heat.
Add the garlic cloves and cook, turning occasionally, until lightly browned, about 3–5 minutes; transfer the garlic to a plate and leave the fat in the pan.
Pat the chicken thighs dry with paper towels and season both sides with salt and pepper.
Increase heat to medium-high and place the chicken in the skillet skin-side down (use a splatter screen if desired). Cook until the skin is browned, about 5–6 minutes, then remove from heat.
Season the chicken with thyme and rosemary (about 1 teaspoon of each per the original directions) and flip so the skin side is up; scatter the previously cooked garlic around the chicken.
Transfer the skillet to the preheated oven and roast until the skin is crispy and the internal temperature reaches 165°F (74°C), about 20 minutes.
Remove the chicken and garlic to a plate. Place the skillet with the drippings over medium heat and bring to a simmer.
In a small bowl, whisk chicken broth with cornstarch until smooth, then whisk this mixture into the skillet drippings and cook, whisking constantly, about 1 minute until slightly thickened.
Remove the pan sauce from heat and stir in lemon juice.
Spoon some pan sauce onto plates, top with a chicken thigh and roasted garlic cloves, and sprinkle with chopped parsley before serving.