In a wide pot or saucepan, melt together the butter and olive oil over medium heat.
Once the butter is melted, add the sliced onions and cook, stirring occasionally, until deeply golden and caramelized, about 30 minutes. Adjust heat between medium and low as needed to caramelize the onions without burning.
When the onions are nearly caramelized, remove the beef patties from the refrigerator and set aside.
Add the mushrooms to the pan with the onions and continue to stir until golden and any liquid from the mushrooms has evaporated. If the pan is too dry, drizzle a little more olive oil.
Stir in garlic and cook until just fragrant, about 30 seconds to 1 minute.
Stir in tomato paste, then the flour.
Slowly pour in the Marsala, scraping the bottom of the pan to remove any browned bits.
Bring the wine to a simmer, stirring often, until the liquid reduces by half and the mixture starts to thicken.
Reduce heat to low, then stir in the cream, parsley, thyme, a pinch of salt, and a pinch of ground black pepper. Adjust flavors to taste and keep over very low heat, stirring occasionally, while the burgers cook.
Heat a grill or cast-iron frying pan over medium-high heat.
If using an indoor grill or cast iron pan, drizzle with a very thin layer of olive oil.
Arrange the ground beef patties in the heated pan or grill, making sure they aren't touching (cook in batches if needed). Grill until browned and the thickest part of the interior is at least 160˚F (71˚C), about 3-4 minutes per side.
During the last 1-2 minutes of cooking, place a provolone slice or two on each burger to gently melt.
Remove the patties from the grill and set aside for 5 minutes to rest briefly before assembling.
If desired, place the burger buns, cut side down, on the grill to toast until golden.
Arrange one or two lettuce slices along the bottom of a burger bun. Top with a cooked patty, then a generous couple of spoonfuls of the warm Marsala mushroom sauce.
Cover with the top half of the burger bun and repeat with the remaining buns and patties.
Serve immediately after assembling.