Preheat the oven to 350 degrees F. Set a rack on a rimmed baking sheet. Toss the mushrooms with the olive oil; season with salt and pepper. Arrange open-side down on the rack. Bake until tender, about 30 minutes.
Meanwhile, melt the butter in a medium skillet over medium heat. Add the garlic and cook until softened, about 2 minutes. Add the panko and cook, stirring occasionally, until golden, about 5 minutes. Stir in the parsley, lemon zest and 1/2 teaspoon salt.
Turn the mushrooms open-side up. Fill each with 1/2 teaspoon of the panko mixture and a cube of cheese; sprinkle with more of the panko mixture. Bake until the cheese is melted and the panko is golden, 5 minutes.