Peel the brown skin from the tops of the mushrooms to reveal the cream colored under skin. Toss the mushroom caps with the 2 tablespoons of the olive oil and season with 1 teaspoon of salt and 1/2 teaspoon of pepper. Place the caps on an aluminum foil lined baking sheet, stem-end up, and roast in the oven until well roasted and browned, about 35 to 40 minutes. Remove from the oven and cool for 10 minutes.
Set a 1-quart saucepan over medium heat and add the butter to the pan. Once the butter is melted, about 1 minute, add the onions and saute until translucent, about 3 minutes. Sprinkle the flour into the pan and stir to form a roux. Continue to cook the roux for 2 to 3 minutes before adding the milk. Bring the milk to a boil and reduce to a simmer, season with the remaining 1/4 teaspoon of salt, 1/4 teaspoon of white pepper and the nutmeg.
Cook the bechamel sauce for 5 minutes before removing from the heat. Allow the mixture to cool slightly before folding in the crabmeat and cheese. Combine the bread crumbs with the remaining tablespoon of olive oil.
Spoon 1/3 cup of the bechamel over each mushroom cap and sprinkle 1 teaspoon of the bread crumb mixture over each stuffed mushroom. Place in the oven and roast until the bechamel is bubbling and lightly browned on the edges, about 8 to 10 minutes. Serve immediately.