Warm a large skillet over medium heat. Melt butter with olive oil.
Add onions and a pinch of the salt. Cook, stirring often, until deep golden, 20–25 minutes.
Stir in garlic and thyme. Cook until fragrant, about 1 minute.
Deglaze with white wine. Scrape up browned bits and simmer until nearly evaporated.
Pour in beef broth and Worcestershire. Simmer until jammy and glazed, 3–5 minutes. Season with remaining salt and pepper.
Mix mayonnaise and Dijon in a small bowl. Spread thinly on one side of each bread slice.
Heat a clean skillet or griddle over medium-low.
Layer Gruyère, a scoop of onions, and provolone on the uncoated side of half the bread. Top with remaining slices, mayo-side out.
Grill sandwiches until golden and melty, 3–4 minutes per side. Cover during the first side to help melt.
Rest 1 minute. Slice and serve hot.