Wash and dry all produce. Adjust rack to middle position and preheat oven to 400 degrees. Peel carrots, then cut on a diagonal into 1/2-inch-thick slices. Toss on a baking sheet with a drizzle of olive oil and a pinch of salt and pepper. Roast in oven until tender, about 25 minutes, flipping halfway through.

Mince 1 clove garlic. Smash other clove with the back of your knife. Peel potatoes and cut into 3/4-inch cubes. Place potatoes in a medium pot with a pinch of salt and enough water to cover by 1 inch. Bring to a boil, then lower heat slightly and let simmer until easily pierced by a knife, 10-12 minutes. Drain.

Heat a drizzle of olive oil in a large pan over medium heat. Season chicken all over with salt and pepper. Add to pan and cook until browned on surface and just shy of being cooked through, 4-5 minutes per side.

Add 1 TBSP butter, smashed garlic, and thyme sprigs to pan. Once butter melts, tilt pan toward you and spoon butter over chicken continuously for 30 seconds. Stir in demi-glace and 1/2 cup water, then increase heat to high. Let bubble until thick and saucy, spooning liquid over chicken every now and then, 2-3 minutes. Remove chicken from pan and set aside to rest.

Add 1 TBSP butter and minced garlic to pot used for potatoes and place over low heat. Once butter melts and garlic is fragrant, about 90 seconds later, return potatoes to pot and pour in 1/3 cup milk (we sent more). Mash with potato masher or fork until smooth. Stir in Parmesan. Season with salt and pepper.

Divide potatoes and carrots between plates, then arrange chicken on top of potatoes. Discard thyme sprigs and garlic clove from pan, then drizzle chicken with remaining sauce in pan. TIP: Stir a splash of water into sauce, if needed, to give it a drizzly consistency.