In a large bowl, toss the mushrooms with the garlic, rosemary, lemon zest and the 1/2 cup of olive oil and let stand for 1 hour.
Light a grill. Spread the mushrooms on a lightly oiled perforated grill tray and season with salt and pepper. Grill over moderately high heat, stirring occasionally, until browned, about 8 minutes. Brush the bread with oil and grill, turning once, until toasted, 1 minute.
Top the toasts with the mushrooms. Top each with a slice of burrata and serve.
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