Preheat the oven to 500°F. Place the rack in the center position.
Wash and dry the fresh produce.
Peel the carrots, then trim off the ends; discard the peels and trimmings. Cut the carrots into ¼-inch-thick rounds (at an angle). Transfer the carrots to a baking sheet.
Trim off and discard the ends of the beets; peel and cut the beets into ¼-inch thick wedges. (To avoid staining, line your cutting board with paper towels before handling the beets.) Transfer the beets to the baking sheet.
One by one, holding a sprig of thyme at its top, run you fingers down it to remove the leaves. Discard the sprigs and chop the leaves. Sprinkle ¾ of the leaves over the carrots and beets (save the rest for garnishing).
Drizzle the carrots and beets with olive oil and season with salt and pepper; toss to coat, then arrange in a single layer.
Place the baking sheet in the oven and roast until the carrots and beets are tender, 15 to 20 minutes.
Preheat a skillet over medium-high heat.
While the skillet heats up, pat the steak dry with paper towels and place on a plate. Season generously with salt and pepper on both sides.
Once the skillet is hot, add coconut oil and swirl to coat the bottom.
Carefully place the steak in the skillet; flipping every 15 to 30 seconds, cook the steak until desired doneness (5 to 7 minutes for medium-rare). (Use a thermometer or cut into the steak to check the doneness.) Once done, transfer the steak to a plate and let it rest for 5 to 10 minutes.
Measure out the feta.
To serve, place the steak on a plate, arrange the roasted vegetables on the side and sprinkle with the feta and remaining thyme. Enjoy!
