Vowelburger™ Inside-Out Burger

^{Inspired by Avi’s Vowelburger™ Riddle here and Ébe Isaac’s Double Supreme Riddle.}

The local Vowelburgers™ has added a new item to their ever-expanding menu, the Inside-Out Burger! This peculiar addition has the two patties on the outside of the burger with the buns sandwiched in between!

Here is the new menu:

Can you identify each type from the menu’s description?