Put the vinegar, water, tamari, yeast flakes, and garlic in a blender or food processor and blend until well combined.
Remove the feed tube and, with the machine running, slowly drizzle in the olive oil.
Blend well until the mixture becomes thick and emulsified, like mayonnaise. Adjust the seasoning with salt as needed.
Pour the mixture into a lidded jar and refrigerate for up to 2 weeks. Bring the sauce to room temperature and shake well before using.
Using your hands, mix the turkey, yeast flakes, Parmesan cheese, umami sauce, salt and pepper in a bowl to combine. Divide the meat into 4 patties. Wash your hands.
Preheat the grill to medium-high heat. Grill the burgers until cooked through, about 5 to 7 minutes on one side then flip over and cook for another 5 minutes on the other side.
Place the burgers on the buns. Top with the grilled onions, roasted mushrooms and drizzle each one with a bit of the Umami sauce.