Heat oil in a skillet over medium heat. Add onion and cook 2-3 mins, until translucent. Add mushrooms and oyster sauce and stir fry another 5-6 minutes, until the moisture disappears.
Remove from heat and mix in the cheese. Stir until well-combined and cheese is melted.
Assemble the sandwiches: sourdough, arugula, mushroom cheese mix, mayonnaise, sourdough. Brush bread with a thin layer of pesto (see notes). Return skillet to medium heat and place the sandwiches pesto side down.
Cook for 3 minutes, or until the bread is evenly golden and crisp, pressing down lightly with a spatula every now and then. Brush on more pesto and flip the sandwiches. Cook for another 3 minutes.
Transfer sandwiches to a cutting board, slice in half and enjoy!
Notes
if you don't want to cook the sandwich in pesto, butter or more olive oil works just the same!