In a small bowl, combine mayonnaise, fresh dandelion flowers, and garlic powder. Set aside.
Caprese Pesto Burgers
In a large bowl, combine ground venison, 2 Tablespoons olive oil, butter, 6 Tablespoons of the basil pesto, salt, and pepper. Gently knead to blend.
Form the mixture into six 4-inch patties, about ¾-inch thick. Brush the burger patties lightly on both sides using 2 Tablespoons of olive oil.
Light a grill and set to medium high heat. Scrape the grill grates well to clean them. When you are ready to grill the pesto burgers, dip a paper towel in remaining 2 Tablespoons of olive oil and grab it with tongs. Wipe down the grill with the oily towel and immediately lay the burgers down on the grill grates and cook for 1 minute.
Turn the heat down to medium, close the grill lid, and grill the burgers for 3 minutes. Flip the burgers, top them with the mozzarella slices and close the grill. Cook for about 3 minutes longer until they are medium rare and reach an internal temperature of 135 degrees F using an instant read meat thermometer.
Grill the buns until lightly toasted, then spread the dandelion flower mayonnaise on the bottoms.
Top the buns with the burgers, the tomatoes and the remaining pesto. Close the burgers and serve immediately.