Heat about 2/3rds of the oil in a large skillet over medium-high heat. Add the onions to one half of the skillet and season with some salt and pepper. Add the veggie patty to the other side. Cook onions until they start to brown, 10-12 minutes. Cook patty 5-6 minutes per side (or time according to package instructions). Turn off heat and set aside.

Stir mayo and mustard together in a small bowl. Spread sauce evenly on all sides of the bread.

Place patty on bottom slice and top with onions and remaining bread slice.

Wipe out the skillet and heat the remaining oil over medium heat. Add the patty melts and cook until golden brown, about 5-7 minutes per side.

Cut in half and serve.