Clean gills from mushrooms. Combine goat cheese, thyme, 1 teaspoon lemon juice mix well. Stuff seasoned goat cheese into mushroom cavity. Grill stuffed mushrooms 10 to 15 minutes, add lemons and bread to the grill. Combine garlic, mayonnaise and remaining 1 teaspoon lemon juice for aioli. Spread aioli generously onto bun. Add arugula, grilled mushroom and grilled lemons to burger.