Heat olive oil and butter together in a skillet over medium-high heat until the butter is melted, 2 to 3 minutes. Saute onion and garlic in the hot oil mixture until the onion begins to caramelize, 15 to 20 minutes.
Pour white wine over the onion mixture; bring to a simmer and cook until fragrant, about 2 minutes. Add beef broth and thyme; cook until the liquid reduces in volume by half. Season the mixture with kosher salt and black pepper.
Trim puff pastry and fit into a tart pan. Score a rim along the edge of the pastry. Puncture the center of the pastry with a fork. Spread the onion mixture over the pastry; top with Parmesan cheese and mozzarella cheese.
Bake in preheated oven until the crust is golden brown, 15 to 20 minutes.