In a food processor, combine the shrimp, egg, cilantro, garlic, 3/4 cup bread crumbs, poblano chile, salt, and pepper. Process until the mixture is a coarse puree. Form the mixture into 6 patties and set aside (if the mixture is too wet, add additional bread crumbs, 1 tablespoon at a time, until the mixture can be molded easily).
Put a grill pan over medium-high heat or preheat a gas or charcoal grill. Brush the patties with vegetable oil and grill until golden, about 3 minutes per side, depending on the thickness.
Scoop the avocado pulp into the bowl of a food processor. Add the mayonnaise, lime juice, garlic cloves, serrano chile and cilantro. Process until smooth. Season with salt and pepper, to taste.
Serve the shrimp burgers on buns, topped with a dollop of avocado aioli, sliced tomato, and lettuce.