Select mushrooms that are fairly level with a wide lip around the edge. Remove the stem.
Heat the grill to medium. Drizzle olive oil on both sides of the mushroom cap and sprinkle with salt and pepper.
Place the mushroom on the grill with the cap side down. Add some butter into the center. Carefully, crack the egg into the well. The mushroom needs to be pretty level or the egg will slide out. Season with salt and pepper.
Close the grill and cook for 10 minutes, or until the yolk is cooked to your preferred doneness.
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