Heat the peanut oil to 350 degrees F in a large cast-iron skillet.
Heat a grill pan over medium-high heat. Grill the pineapple slices to get nice grill marks, about 2 minutes per side. Butter and grill the buns as well.
Set up a breading station of flour, eggs and panko seasoned with salt and pepper, in 3 separate shallow dishes. Dredge each mushroom cap in the flour first, then the egg and finally the panko. Fry the breaded mushrooms until golden brown and cooked through, 4 to 5 minutes. Drain on a baking sheet lined with a cooling rack or paper towels.
Whisk together the mayonnaise and teriyaki sauce in a small bowl.
For the sandwich build: Spread some teriyaki mayo on the bottom and top halves of the buns. Top the bottom bun with a pineapple slice, a fried portobello mushroom cap, some onion slices and red lettuce leaf. Oh man!