Preheat a grill to medium high. Put the asparagus and mushrooms in separate bowls; toss each with 1 tablespoon olive oil, 1 garlic clove, 1 teaspoon thyme and a pinch each of salt and pepper. Place the asparagus and mushrooms on opposite sides of a grill basket and grill, tossing occasionally, until tender and browned around the edges, about 3 minutes for the asparagus and 7 minutes for the mushrooms. Remove to a large plate.
Heat a large cast-iron skillet on the grill; add the remaining 3 tablespoons olive oil. Crack the eggs into a bowl, then pour into the skillet; season with salt. Scatter the vegetables around the egg whites, avoiding the yolks. Cook, rotating the skillet, until the whites are set but the yolks are still runny, 5 to 8 minutes. Meanwhile, grill the bread until lightly charred, 1 minute per side.
Remove the skillet from the grill and sprinkle with parsley, parmesan and red pepper flakes. Serve with the grilled bread.