Wash and dry all produce. Adjust rack to top position and preheat oven to 450 degrees. Pick and chop 1/2 tsp thyme leaves from sprigs. Trim and thinly slice scallions, separating greens and whites. Zest 1 tsp zest from lemon, then quarter. Place peppercorns in a zip-close bag; crush with a rolling pin or a heavy pan.

Toss carrots on a baking sheet with a drizzle of olive oil, salt, and pepper. Roast until tender, about 15 minutes, tossing halfway. Meanwhile, halve potatoes lengthwise. Place in a large, microwave-safe bowl and cover with plastic wrap. Microwave on high for 7 minutes; keep in microwave until step 6.

Melt garlic herb butter in a small pot over medium-high heat. Whisk in flour and let cook 30 seconds. Pour in milk and whisk until smooth. Add a few whole thyme sprigs (use the rest as you like), scallion whites, and large pinch of salt to pot. Bring to a simmer and let bubble until thickened, about 2 minutes.

Remove pot with sauce from heat, then stir in Gruyère cheese and half the Parmesan. Season with salt and pepper. Cover and set aside. Pat steak dry with a paper towel. Season all over with plenty of salt and crushed peppercorns (save some pepper for later). Heat a drizzle oil in a large pan over medium-high heat. Add steak and cook to desired doneness, 3-5 minutes per side. Set aside on a cutting board. Reduce heat under pan to medium.

Once carrots are done, toss with lemon zest and a few squeezes of lemon juice. Preheat broiler to high. Add 1/2 cup water and demi-glace to pan used for steak over medium heat. Cook until just thickened, about 2 minutes. Remove from heat, stir in 2 TBSP plain butter, and season with salt and crushed peppercorns. Meanwhile, in a small bowl, mix panko, remaining Parmesan, chopped thyme, a large drizzle olive oil, and a pinch of salt and pepper.

Drain any water from potatoes, then place in a baking dish. Season with salt and pepper, then pour cheese sauce over top. Sprinkle crust over. Broil until golden brown, about 3 minutes. Divide steak between plates (slice first if desired). Drizzle with gravy. Serve with carrots and potatoes on the side. Garnish everything with scallion greens.