Heat the olive oil in a large skillet on medium-high heat for 5 minutes.
Hand wipe the mushrooms with a damp paper towel to clean them.
Using a food processor, grind the mushrooms for 12-15 seconds to turn them into a minced texture.
Add the minced mushrooms to the skillet and spread them out. Cook for 8 minutes, stirring at the halfway point.
Chop the shallots and garlic cloves in the food processor for 5-6 seconds to mince them.
Add the shallots and garlic to the skillet with the mushrooms. Stir frequently while cooking for 2-3 minutes. Make sure the mixture is dry. Season with a 3 finger pinch of kosher salt. Remove from the heat to let cool.
While the mushroom mixture is cooling, add the ground turkey, Greek yogurt, breadcrumbs, egg, salt, and pepper to a mixing bowl. Add the mushroom mixture to the bowl.
Mix all the ingredients with your hands or a spatula.
Divide the mixture into quarters. Patty the burger mix to 1-inch thickness. You will have 4 patties total.
Place on a parchment-lined cookie sheet or plate. Chill in the fridge for 20 minutes.