Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Trim and peel carrots, then cut on a diagonal into 1-inch-thick pieces. Trim and thinly slice mushrooms. Trim and thinly slice scallions, separating whites from greens.

Toss carrots on a baking sheet with a drizzle of oil, salt, and pepper. Roast until tender, 25-30 minutes. Meanwhile, dice potatoes into 1/2-inch pieces (peel first for a smoother texture). Place in a large pot with enough salted water to cover by 2 inches. Boil until very tender, 15-20 minutes. Reserve 1/4 cup potato cooking liquid (1/2 cup for 4); drain and return potatoes to pot.

Meanwhile, pat chicken dry with paper towels; season all over with salt and pepper. Heat a drizzle of oil in a large pan over medium-high heat. Add chicken and cook until browned and cooked through, 5-7 minutes per side. Turn off heat; transfer to a plate.

Heat pot with potatoes over low heat; add 1 TBSP butter (2 TBSP for 4 servings) and half the sour cream. Using a potato masher or fork, mash potatoes, adding splashes of reserved potato cooking liquid as needed, until smooth and creamy. Season with salt and pepper.

Heat a drizzle of oil in pan used for chicken over medium-high heat. Add mushrooms and scallion whites; season with salt and pepper. Cook, stirring, until softened, 2-4 minutes. Stir in stock concentrate and 1/4 cup water (1/3 cup for 4 servings). Bring to a boil, then reduce to a simmer. Cook until slightly thickened, 1-2 minutes. Turn off heat. Stir in remaining sour cream and 1 TBSP butter. Season with salt and pepper.

Add chicken to same pan over medium-low heat; spoon some sauce over. Evenly top chicken with cheese. Cover pan until cheese is melted, 1-2 minutes. Divide carrots, potatoes, and chicken between plates. Spoon remaining sauce over chicken and potatoes. Garnish with scallion greens and serve.