Melt 2 tablespoons of unsalted butter in a large nonstick skillet over medium-low heat.

Add the sliced yellow onions and cook for 15 minutes, stirring occasionally, until they begin to caramelize.

Add the sliced mushrooms and continue to cook until the mushrooms are tender and the onions are brown and caramelized, about 10 more minutes.

Season the caramelized onions and mushrooms with salt and black pepper to taste.

Transfer the onions and mushrooms to a bowl and set aside.

Place the skillet back on the stove and turn the heat to medium.

Ensure there is enough fat in the pan to cook the veggie patties; otherwise, spray the pan with nonstick spray.

Once the pan is hot, add the veggie burgers.

Cook the veggie burgers according to package instructions, usually about 6-7 minutes per side, or until completely cooked through.

Remove the veggie burgers from the heat.

Assemble each sandwich by placing a slice of provolone cheese on the bottom slice of rye bread.

Top each sandwich with a handful of baby arugula and then place a cooked veggie burger on top of the arugula.

Top the veggie burgers with the caramelized onions and mushrooms, and then place another slice of provolone cheese on top.

Place the remaining slice of rye bread on top of the provolone cheese to complete the sandwich.

Preheat a griddle pan or a large nonstick skillet over medium heat.

Add the remaining 2 tablespoons of butter to the hot griddle or skillet.

Cook the sandwiches in batches, being careful not to overcrowd the pan.

Cook the sandwiches for about 3-4 minutes per side, or until they are golden brown on each side, adding more butter if needed.

Serve the patty melts immediately and enjoy!