Make the cilantro-lime sauce by adding all of the ingredients to a food processor or blender and blending until mixed.
Place the tilapia in a food processor and pulse until chopped. If you don't have a food processor, you can cut the fish into small pieces with a sharp knife or kitchen scissors, but you may need more breadcrumbs to bind the patties.
Transfer the tilapia to a large bowl, add the breadcrumbs, egg, garlic, cilantro, paprika, salt, and pepper, and mix well. Roll a small amount into a ball to test it -- if it doesn't stick together well, add a little more breadcrumb until it does.
Form the mixture into 4 large patties, pressing them tightly into shape.
Heat the oil in a large skillet -- preferably cast-iron -- on medium-high heat.
Place the patties in the pan and cook for 4–5 minutes on one side, then flip them and cook for 3–4 minutes on the other side until both sides are browned and crispy. Do this in batches if you have a small pan.
Place one lettuce leaf on the bottom of each hamburger bun, then top each with a burger. Spread cilantro-lime sauce on top.
Top the burgers with thinly sliced radish and sliced avocado.