Heat the olive oil in a skillet over medium-high heat and cook the mushrooms for 8 to 10 minutes until deeply browned and dry.
Pulse the lentils and oats in a food processor 5 times until coarse, not smooth.
Mix the lentil mixture with the mushrooms, egg, soy sauce and salt, then cool it for 10 minutes and shape 4 compact patties.
Stir the yogurt, garlic, parsley and lemon juice together and refrigerate the sauce.
Cook the patties in a lightly oiled nonstick skillet over medium heat for 5 to 6 minutes per side until browned and firm.
Rest them for 3 minutes and toast the buns in the cleared skillet.
Spread garlic herb yogurt on both bun halves and assemble each with arugula and one lentil mushroom patty.