Heat skillet over medium-high heat with 1 tablespoon vegetable oil.
Season steak with salt and pepper to taste on both sides, then place into hot skillet and cook for about 4-6 minutes until browned. Flip steak and continue to cook until browned on the second side and an internal temperature of 145°F. (for medium) about 5 more minutes.
Remove from skillet and allow to rest on a cutting board until ready to serve at least 5 minutes.
While steak is resting, use a paper towel to wipe skillet clean.
Heat skillet over high heat with 1 tablespoon vegetable oil.
Add carrots and cook until dark brown and charred, stirring frequently, about 5 minutes total cook time.
Hold one end of thyme stem and use other hand to strip about 1 teaspoon of leaves into skillet (about 2-3 full stems)
Toss in garlic and cook about 1-2 minutes. Remove from heat leaving carrots in skillet.
In a small mixing bowl, mix olive oil and vinegar and season to taste with salt and pepper to make a simple vinaigrette.
Pour half the vinaigrette over burnt carrots and toss to coat evenly. Cut open one end of bagged arugula and pour remaining vinaigrette into gently dress the greens.
Slice rested steak across the grain into 1/2" slices.
Divide between 2 plates and spoon half the chimichurri over each portion of steak.
Split arugula between the plates and top with burnt carrots. Sprinkle crumbled goat cheese over carrots before serving.