Add 1 tablespoon of olive oil to a large non-stick pan or wok over medium-high heat. When the oil begins to shimmer (around 1 minute), add the julienned onion. Cook for about 3 minutes, and then add salt, until they just start to gain color. Add apples and cook for an additional 3 to 4 minutes, until they start to brown. Add 1/4 cup of water and cook until all the liquid has evaporated. Continue this process for about 3 more minutes, until apples and onions are a dark brown. Set aside.
Butter one side of each slice of bread and set aside. Whisk together vegan mayo, mustard, and vegan parmesan. Set aside.
Add 1 tablespoon of olive oil to a large non-stick pan or griddle over medium-low heat. Place 2 slices of bread, butter side down, onto the pan. Slather mayo mixture onto each slice of bread (reserving enough for second sandwich). Spoon half the apple onion mixture onto one slice of bread. Place provolone slices and shredded cheddar on the other slice of bread.
Add 1/4 cup of water to the pan and quickly pop the lid. Cook for about 3 minutes, until cheese looks like it's melted. Flip one slice of bread on top of the other. Press down with spatula gently, make sure the filling doesn't spill out. Repeat with remaining ingredients for second sandwich.