Step 1: Preheat your oven to 400°F (200°C). Toss the sliced carrots with olive oil, salt, and pepper on a baking sheet. Roast for 20-25 minutes until tender and caramelized.
Step 2: Bring a large pot of salted water to a boil. Cook the linguine according to package instructions until al dente. Reserve 1/2 cup of pasta water before draining.
Step 3: Season the steak generously with salt and pepper. In a large skillet over medium-high heat, sear the steak for 3-4 minutes per side (for medium-rare). Remove from the pan and let it rest for 5-10 minutes before slicing into thin strips.
Step 4: In the same skillet, reduce heat to medium and melt the butter. Add the garlic and cook for 1 minute until fragrant. Whisk in the Dijon mustard, lemon juice, lemon zest, red pepper flakes, parsley, and chives.
Step 5: Add the cooked linguine and roasted carrots to the skillet with the cowboy butter sauce. Toss well to coat, adding reserved pasta water if needed to loosen the sauce. Top with the sliced steak and serve immediately.