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

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. Boil until tender when pierced with a fork, about 15 minutes. Drain and return to pot. Mash with 1 TBSP butter and 1/4 cup milk. (TIP: Add more milk, as necessary, until smooth and creamy.) Stir in Parmesan and 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 on top and flecked with amber brown bits, 1-2 minutes. Turn off heat and 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 to pan 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 and cook until softened, 1 minute. Stir in demi-glace, 1/4 cup water (1/3 cup for 4 servings), 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 to thoroughly coat. Divide veggies, potatoes, and beef between plates. Top beef with sauce and remaining parsley.