In a large nonstick or enamel-coated skillet over medium-high heat, heat 1 tablespoon of olive oil. When the oil is shimmering, add the mushrooms and sauté until tender and mushrooms release their water, 7-10 minutes.
Stir the garlic, rosemary, and red wine into the skillet and sauté 1 minute more, stirring up any bits stuck to the bottom of the skillet. Season with salt and pepper and remove from heat.
To serve, spread burrata over toasts or crackers. Spoon the sautéed mushroom mixture over top. Alternatively, you can serve by placing the burrata ball on a tray and top with the mushroom mixture; drizzle with a little more olive oil. Serve with toasts or crackers so guests can serve themselves.
