Preheat oven to 400°F. Remove stems from portobello caps and scrape out the gills.
Brush caps with olive oil, season with salt and pepper, and roast gill side down for 10 minutes. Flip and roast 5 more minutes.
Combine crab meat, panko, egg, onion, garlic, lemon juice, half the cheese, thyme, salt, and pepper in a bowl. Mix gently.
Drain any liquid from the mushroom caps, then divide the filling evenly among them.
Top with remaining cheese.
Bake for 18 to 20 minutes, until golden and bubbly. Broil for the last 1 to 2 minutes if extra browning is desired.
Let rest 5 minutes before serving.