Preheat oven to 350 F degrees. Season chicken thighs on both sides with salt and pepper.
Add 1 Tbsp butter and 1 Tbsp olive oil in a large cast iron, or oven-safe skillet, and heat over MED-HIGH heat. Add chicken thighs to pan, working in batches if needed to avoid overcrowding, and cook 5 minutes per side, until golden brown.
Transfer chicken to plate. Add garlic cloves to the pan and cook 2-3 minutes, until lightly golden and soft. Be sure to stir often to keep the garlic from burning. Transfer garlic to plate with the chicken.
Add white wine and scrape the bottom of the pan to loosen any browned bits. Bring to a simmer and cook about 2 minutes, until wine reduces by about half.
Add chicken stock and remaining 1 Tbsp butter. Return to a simmer. Season with salt and pepper, to taste, and turn off heat.
To a mixing bowl, combine potatoes, carrots and remaining 1 Tbsp olive oil, stirring to coat.
Transfer chicken and garlic cloves back to the pan, spooning some of the sauce over the top of the chicken. Nestle potatoes and carrots around the chicken thighs. Top with thyme sprigs.
Bake 35-45 minutes, until chicken is fully cooked and potatoes and carrots are fork tender.