Thinly slice onion. Heat 1 tablespoon each of butter and oil in a medium ovenproof skillet over medium-high. Add sliced onions, 1 teaspoon sugar, and a pinch each of salt and pepper. Cook, stirring, until well browned, adding 1 tablespoon water at a time, and scraping up any browned bits, 12–15 minutes. Transfer onions to a bowl. Wipe out skillet; reserve for step 3.

Meanwhile, thinly slice 1 large garlic clove; reserve a second whole garlic clove for step 5. Halve rolls, drizzle cut sides with oil, then season with salt and pepper. Pick and finely chop 2 teaspoons thyme leaves; discard stems.

Preheat broiler with a rack in the top position. Pat chicken dry and season all over with salt and pepper. Heat 2 teaspoons oil in reserved skillet over medium-high. Add chicken and cook, until well browned and cooked through, 3–4 minutes per side. Transfer to a plate.

Add sliced garlic and 1 teaspoon oil to same skillet. Cook, stirring, until garlic is lightly golden, about 30 seconds. Add chopped thyme, broth concentrate, and ½ cup water: bring to a simmer. Cook, scraping up any browned bits, until sauce is slightly thickened, 1–2 minutes. Return chicken to skillet and top with caramelized onions and cheese.

Transfer skillet to top oven rack and broil until cheese is melted and golden brown, 2–4 minutes (watch closely as broilers vary). Carefully, remove skillet from oven. Broil rolls directly on top oven rack until golden brown and toasted, 1–2 minutes per side (watch closely). Remove rolls from oven and rub cut sides with reserved whole garlic clove.

In a medium bowl, whisk to combine 2 tablespoons oil, 1 tablespoon vinegar, 1½ teaspoons Dijon mustard, and a pinch of sugar. Season to taste with salt and pepper. Add spinach; toss to coat. Serve chicken with salad and garlic bread alongside. Enjoy!