Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Dice potatoes into 1/2-inch pieces. Trim, peel, and cut carrots and parsnip on a diagonal into 1/2-inch-thick pieces. Halve, peel, and mince shallot until you have 2 TBSP (4 TBSP for 4 servings). Finely chop parsley leaves and stems.

Toss carrots and parsnip on a baking sheet with a large drizzle of olive oil and a pinch of salt and pepper. Roast on top rack until golden brown and tender, 20-25 minutes.

Meanwhile, place potatoes in a medium pot with enough salted water to cover by 2 inches. Bring to a boil and cook until tender, 15-20 minutes. Drain and return potatoes to pot. Mash with 1 TBSP butter until smooth and creamy, adding splashes of milk as needed. Stir in Parmesan. Season with salt and pepper. Cover to keep warm.

While potatoes cook, melt 2 TBSP butter (3 TBSP for 4 servings) in a medium, preferably nonstick, pan over medium-high heat. Cook, stirring frequently, until butter is foamy and flecked with amber brown bits, 1-2 minutes. Turn off heat and carefully transfer to a large bowl. Wipe out pan.

Pat beef dry with paper towels and season all over with salt and pepper. Heat a drizzle of oil in pan used for butter over medium-high heat. Add beef and cook to desired doneness, 4-7 minutes per side. Turn off heat; transfer to a plate to rest. Wash out pan.

Melt 1 TBSP plain butter in same pan over medium-low heat. Add minced shallot; cook until softened, 1 minute. Stir in demi-glace, 1/4 cup water (1/3 cup for 4), and any resting juices from beef. Simmer until thickened, 2-3 minutes. Turn off heat. Stir in 1 TBSP plain butter; season with salt and pepper. Add carrots, parsnip, and half the parsley to bowl with brown butter; toss until thoroughly coated. Divide veggies, potatoes, and beef between plates. Top beef with sauce and remaining parsley. Serve.