Sauté onion in olive oil in a large skillet over medium-high heat 5 minutes. Cover, reduce heat to medium, and cook 15 to 20 more minutes or until onion is tender, stirring often. Uncover and cook, stirring constantly, 5 to 10 minutes or until browned. Remove from heat.
Grease 10-inch cast-iron skillet with butter. Layer half of turnip, potato, and apple slices in skillet; sprinkle with half the onion, rosemary, thyme, sea salt, pepper, and cheese. Repeat procedure.
Bake, uncovered, 40 minutes or until tender. Let stand 15 minutes. Garnish, if desired.