Make the garlic butter: Pulse the garlic in a food processor until chopped. Add the butter, lemon zest, lemon juice, parsley, 1 teaspoon salt and 1/2 teaspoon pepper; pulse until smooth. Chill at least 30 minutes.
Prepare the mushrooms: Preheat the oven to 400 degrees F. Toss the mushrooms with 1 tablespoon olive oil and season with salt. Fill each mushroom with 1 heaping teaspoon of the garlic butter (reserve the remaining garlic butter for the crab filling). Arrange the mushrooms snugly in a baking dish and bake until tender, about 15 minutes.
Meanwhile, heat the remaining 1 tablespoon olive oil in a skillet over medium heat. Add the shallot and crabmeat; cook until the shallot is soft, about 3 minutes. Stir in the wine and cook until slightly reduced, about 1 minute. Stir in the reserved garlic butter (about 1/3 cup) until melted. Season with salt and pepper. Remove from the heat and push the crab mixture to one side of the skillet; let any excess juices collect.
Remove the mushrooms from the oven and preheat the broiler. Stuff the mushrooms with the crab mixture. Toss the panko with the juices in the skillet; sprinkle over the mushrooms. Broil until golden, about 2 minutes. Squeeze lemon wedges on top.