In a large skillet, melt butter over medium-low heat. Add sliced onions and cook slowly, stirring often, until golden brown and caramelized (20–25 minutes).
Stir in garlic and cook 1 minute until fragrant.
Add rice and stir to coat with onion-butter mixture; cook 2–3 minutes until slightly toasted.
Pour in broth, add thyme, and season with salt and pepper. Bring to a boil.
Reduce heat to low, cover, and simmer until rice is tender and liquid absorbed (18–20 minutes).
Remove from heat, fluff with a fork, and stir in an extra pat of butter.
Sprinkle cheese over the hot rice, cover for 2 minutes, and let it melt through (Optional).
Garnish with parsley and serve warm.