In a medium saucepan, combine the peaches, butter and brown sugar. Cook until the butter has melted and the peaches start to break down. Use a potato masher to mash the peaches. Cook the sauce over medium heat until slightly thickened, 10 to 15 minutes.

Once thickened, remove from heat and stir in basil. If you want a stronger basil flavor, use 1 tablespoon, but if you want the basil flavor more muted, only use 1/2 tablespoon.

Preheat the oven to 400F. Place a baking rack on top of a baking sheet and set aside.

In a shallow bowl, combine the cornstarch, salt, paprika and cayenne. Put the egg whites in another shallow bowl. Place the coconut in a third shallow bowl. Holding a shrimp by the tail, dip it in the cornstarch mixture and coat the shrimp. Tap off any excess. Dip into the egg white, then into the coconut. Use your fingers to press the coconut into the shrimp. Place on the prepared baking sheet. Repeat with the remaining shrimp.

Bake in the preheated oven until slightly browned, about 15 to 20 minutes, flipping over once. Serve with the basil-peach sauce.