Puree the tails and a small handful of the chopped pieces into a small food processor (I use the unit that came with my stick blender). Place the remaining pieces into a medium bowl.
Season to taste with salt, pepper, Old Bay and chopped fresh herbs as desired and form into patties. If the patties don't stick together well you can always puree another small handful of the mix to help hold them together. I got four burger sized patties about an inch thick but you can make them smaller for sliders.
Heat the butter and olive oil in a large skillet. When very hot, carefully transfer the patties to the skillet and fry the first side for about 2 minutes. You'll see the color change on the sides of the patties as the shrimp cooks.
Carefully turn the shrimp burger to the other side, reduce the heat to medium low and cover the pan with a lid. Cook until both sides are golden brown and the shrimp are just cooked through, about 3 minutes more.
Drain briefly on paper towels and serve immediately with the garlic red pepper mayo.