Preheat your oven to 425°F (220°C). Peel and slice carrots, toss with olive oil, salt, and pepper, spread on baking sheet, and roast for 15-20 minutes.

Bring salted water to boil in a large pot, add linguine, and cook for 8-10 minutes. Reserve one cup of pasta water, drain, and set aside.

Heat olive oil in a skillet over medium-high heat. Season steak with salt and pepper, sear for 2-3 minutes each side for medium-rare, then rest on a cutting board.

In the same skillet, reduce heat and add butter and remaining oil until melted. Add garlic and sauté for 1 minute. Stir in lemon juice, Dijon mustard, and red pepper flakes.

Add cooked linguine to the sauce, tossing to coat. Add sliced steak and roasted carrots, mixing gently, adjust sauce with reserved pasta water if needed.

Serve warm, garnished with Parmesan cheese and parsley. Adjust seasoning as necessary.