How to Make It
Prepare the Hamburgers: Gently mix together the ground beef and next 5 ingredients (through onion powder) in a bowl. Shape mixture into 3 (1/2-inch-thick) patties. Heat a large cast-iron skillet or grill pan over medium-hight. Coat with cooking spray. Cook patties to desired degree of doneness, about 2 to 3 minutes per side for medium. Remove from skillet; let stand until cool enough to handle.
Prepare the Caramelized Onions: Melt butter in a skillet over high until very foamy. Add onion; cook, stirring often, until beginning to brown, about 3 minutes. Reduce heat to medium, and sprinkle with salt. Cook, stirring occasionally, until deep brown and soft (do not cook until mushy), 8 to 10 minutes. Remove from heat. Caramelized Onions can be stored in an airtight container in the refrigerator for up to 3 days.
Prepare the Omelet: Break 1 Hamburger into 1/2-inch chunks. Reserve remaining 2 Hamburgers for additional Omelets or another use. Whisk together eggs, water, salt, and pepper in a bowl until smooth.
Melt butter in a nonstick skillet over medium until bubbly, about 1 minute and 30 seconds. Pour egg mixture into skillet. Cook until edges are set and top just begins to look dry, about 2 minutes, occasionally lifting edges with a rubber spatula to allow uncooked egg to drip to bottom of skillet. Place Hamburger chunks and 2 tablespoons of the Caramelized Onions over half of the Omelet. Place 3 cheese triangles on filling; reserve remaining cheese triangle for another use. Cover and cook until cheese melts, about 30 seconds. Uncover and place bacon pieces and tomato on cheese. Fold Omelet in half to cover filling.
Slide Omelet from skillet onto a plate. Top evenly with potato sticks and ketchup.