Heat a large skillet over medium heat. Add the butter pats and oil. When the butter has melted, stir in the mushrooms and garlic and season with salt and pepper. Cook for 12 to 15 minutes, until all of the liquid evaporates. Set aside to cool.
In a medium sized bowl, mix together the parsley, egg, breadcrumbs, rice and cheese. Stir in mushrooms and mix until well combined. Form into 4 patties and place in the refrigerator for 30 minutes.
Cook in a skillet for 8 minutes per side, or until golden and crisp, pressing down gently with a spatula as they cook.
To assemble: Place a burger on each toasted bun with a slice of tomato, lettuce and mustard.