1. Preheat oven and roast carrots: Preheat oven to 400 degrees. Mince 1 clove garlic, smash remaining clove. Roast the peeled carrots into ½-inch coins tossed on a baking sheet with a drizzle of olive oil and a pinch of salt and pepper until softened (25 minutes).
2. Prep and Boil Potatoes: Place peeled and cut potatoes into ¾-inch cubes in a medium pot with a pinch of salt and enough water to cover. Bring to a boil, after that drain them.
3. Cook chicken: Season chicken on all sides with salt and pepper. Sear until golden brown and almost cooked.
4. Baste chicken and make glaze: Add 1 TBSP butter, smashed garlic, and thyme sprigs to pan. Once butter melts, tilt pan towards you and repeatedly spoon infused butter over chicken for about 30 seconds. Add demi-glace and ½ cup water, and increase heat to high. Stir to combine, then reduce until thick. Spoon glaze over chicken. Remove from pan and set aside to rest.
5. Mash potatoes: We all know how to that. Add milk, mash in Parmesan cheese, and season with salt and lots of pepper.
6 Plate: Serve chicken on a bed of mashed potatoes and carrots.