Preheat oven to 425°F and line a large rimmed baking sheet with parchment.
Cut carrots into sticks and slice apples into wedges; set apples aside for later.
Whisk olive oil, maple syrup, Dijon, vinegar, salt, pepper, thyme, and red pepper flakes in a large bowl.
Toss carrots with about two-thirds of the glaze. Spread on the sheet pan in a single layer.
Roast 15 minutes, stirring once halfway.
Coat apples with remaining glaze. Scatter apples over the carrots and toss lightly on the pan.
Roast 8–10 minutes more until carrots are tender and edges caramelize. Broil 1–2 minutes for extra color if desired.
Transfer to a warm platter and sprinkle with a few extra thyme leaves. Serve right away.