Preheat your Le Griddle flap-top grill. Turn one half to high heat, and the other to low.
Combine all ingredients for the burger patty in a medium bowl. Mix by hand, just until everything is evenly distributed. Remove a couple of tablespoons of the mixture and reserve for the gravy. Form the rest into 6 equal sized balls (about 1/3 lb each). Store in the refrigerator, covered in plastic, until ready to cook.
Place a Lodge 8” Cast Iron Griddle on the hot side of the griddle. To make the gravy, add the reserved tablespoons of the meat mixture to the skillet to brown. Then, melt the butter in the skillet. Whisk in the flour and Cattleman’s Grill Trail Dust All Purpose Seasoning. Cook until toasted, about 60 seconds, whisking frequently. Slowly whisk in the half and half until it is all incorporated. Bring to a simmer and allow to thicken. Remove from the griddle.
Using a large metal spatula, smash each portion of burger into the hot side of the griddle surface, pressing until the burger is very thin and wider than the diameter of the buns.
At the same time, spray the cooler half of the grill with Spray Duck Fat. Place the cut sides of the burger buns on the griddle to toast.
Flip the burgers when seared and browned. Immediately top with shredded cheddar cheese to melt into the burger patties. Spoon a couple of tablespoons of the cream gravy over each patty. When the bottoms are browned and cheese is melted, stack the cheese and gravy topped patties two high. Transfer the patties to the brioche buns (two patties per bun).
Slide the burgers to the edge of the griddle.
Spray a wide-mouth mason jar ring/lid with the duck fat. Spray the cooler side of the griddle with duck fat. Place the ring on the griddle, Crack an egg into the ring and press until the egg begins to set in a perfect circle shape. Remove the ring and cook until the whites are set, but yolks are still runny.
Place the eggs on top of the burger patties and cover with the top bun. Serve the B&G Breakfast Burgers topped with Cattleman’s Grill Pit Fire Hot Sauce.