Set oven to 400°F and line a baking sheet with parchment paper.
In a food processor or with a knife, finely chop almonds until they resemble coarse breadcrumbs. Mix with grated parmesan and a pinch of salt and pepper in a shallow bowl.
In a separate bowl, beat the eggs. Pat mushrooms dry and dip each one first into the egg, then roll in the almond-parmesan mixture, pressing gently to coat.
Arrange coated mushrooms on the prepared baking sheet. Bake for 18–20 minutes, flipping halfway through, until golden and crisp.
In a small bowl, stir together mayonnaise, capers, pickles, onion, and a pinch of salt and pepper.
Plate the crispy mushrooms with the dipping sauce on the side.