Sauté the Mushrooms: Heat 1 tablespoon of olive oil in a large nonstick or enamel coated skillet over medium high heat. Once the oil is shimmering, add the chopped mushrooms.
Cook, stirring occasionally, for 7-10 minutes, or until the mushrooms have released their liquid, the liquid has cooked off, and the mushrooms are tender and starting to turn golden brown.
Add Aromatics and Deglaze: Add the minced garlic, rosemary, and red wine to the skillet. Stir constantly, scraping up any browned bits from the bottom of the skillet, and cook for 1 minute more until fragrant.
Season the mushroom mixture with the sea salt and black pepper. Remove the skillet from the heat.
Assemble and Serve: To serve, spread a generous amount of burrata cheese over each piece of toasted sourdough bread or cracker. Spoon the warm sautéed mushroom mixture over the top.
Alternatively, for family style serving, place the whole burrata ball on a serving platter. Spoon the warm mushroom mixture directly over the top of the cheese. Drizzle everything with a little more extra virgin olive oil. Serve immediately with toasts or crackers on the side for guests to serve themselves.