Trim bottoms off Brussels sprouts and halve vertically (quarter if larger than ping pong balls). Peel, trim, and cut carrot into 1/4" diagonal slices. Stem thyme and coarsely chop. Rinse pork tenderloin, pat dry, and season with 1/4 tsp. salt and 1/4 tsp. pepper.

Place carrots and Brussels sprouts onto prepared baking sheet. Toss with 1 Tbsp. olive oil, half the Parmesan (reserve remaining for cooked vegetables), 1/2 tsp. salt, and 1/4 tsp. pepper and spread into a single layer on one half of baking sheet. Roast 5 minutes and remove from oven. Vegetables will finish cooking in a later step.

Place a medium pan over medium-high heat. Add 2 tsp. olive oil and pork to hot pan. Sear until browned on three sides, 2-3 minutes per side. Remove to other half of baking sheet with partially roasted vegetables. Reserve pan; no need to wipe clean.

Return baking sheet to oven and roast until vegetables are browned and tender and pork reaches a minimum internal temperature of 145 degrees, 8-12 minutes. Remove from oven and sprinkle remaining Parmesan and half the thyme (reserve remaining for sauce) onto vegetables. Let pork rest 5 minutes. While pork rests, make cider beurre blanc.

Return pan used to sear pork to medium-high heat. Add apple cider, bring to a boil, and reduce by half, 1-3 minutes. Remove from burner, swirl in butter, and add remaining thyme. Season with 1/4 tsp. salt and a pinch of pepper.

Slice pork tenderloin into 1/2" pieces. Place roasted vegetables onto plate. Add sliced pork in front of vegetables. Spoon cider beurre blanc in front of pork.