Step 1: Preheat your oven to 400°F (200°C). Wash and peel your carrots, then cut them into uniform 1-inch pieces. Toss them with 2 tablespoons of olive oil, salt, and pepper on a baking sheet. Spread them in a single layer. Roast for 20-25 minutes, or until tender and slightly caramelized, flipping halfway through. Set aside.
Step 2: Bring a large pot of generously salted water to a rolling boil. Add the linguine and cook according to package directions until al dente. Reserve about 1 cup of pasta water before draining.
Step 3: Pat your steaks thoroughly dry with paper towels. Season generously with salt and black pepper on both sides. Heat 1 tablespoon of olive oil in a heavy-bottomed skillet (cast iron works wonderfully) over medium-high heat until shimmering and almost smoking. Carefully place the steaks in the hot pan. Sear for 3-5 minutes per side for medium-rare, or until your desired doneness, developing a deep brown crust. Remove steaks from the pan, tent with foil, and let rest for at least 5-10 minutes.
Step 4: In the same skillet (no need to clean it!), reduce heat to medium-low. Add the unsalted butter and let it melt. Stir in the minced garlic, fresh parsley, chives, Dijon mustard, red pepper flakes, Worcestershire sauce, and a squeeze of lemon juice. Cook gently for 1-2 minutes, just until fragrant, being careful not to burn the garlic. Season with salt and pepper to taste.
Step 5: Slice the rested steak against the grain into thin strips. Add the drained linguine directly into the skillet with the cowboy butter. Toss to coat evenly. If the sauce seems too thick, add a splash or two of the reserved pasta water until it reaches your desired consistency. Gently fold in the roasted carrots and sliced steak. Serve immediately, garnished with extra fresh parsley or chives, if desired.