Heat oven to 400°. Melt 2 tbsp. butter in a 12″ ovenproof skillet over medium-high. Working in batches, cook turnips, cut side down, until browned, 4–6 minutes. Return all turnips to pan and season with salt. Transfer to oven; bake until tender, 12–15 minutes. Transfer to a serving platter; keep warm.
Wipe skillet clean and toast caraway seeds over medium-high until they pop, 1–2 minutes; transfer to a bowl. Add remaining butter to skillet; melt. Cook turnip greens until wilted, 3–4 minutes. Stir in reserved caraway seeds and salt; cook 1 minute. Transfer to platter with roasted turnips.