Divide the flour evenly between two small bowls. Crack eggs into a separate bowl and whisk until combined.
Dredge the onions in the flour, then the eggs, then the second batch of flour.
Shake the excess dredge off the onions and deep fry at 350⁰F for about 2 minutes, or until golden brown and crispy. Remove from fryer and allow to drain on paper towels.
Grill the INCOGMEATO Burger Patty over medium-high heat for about 4 minutes on one side. Flip the patty, cover with cheese, and continue to cook until the cheese is melted and the patty is cooked through, about 4 minutes. Slice brioche bun and toast until golden brown on the inside.
To assemble, spread half of the barbecue sauce and half of the mayonnaise on the bottom bun, add the patty, and then top with the fried onions. Spread remaining barbecue sauce and mayonnaise on the top bun and top the burger.