Preheat oven to 375°F.
Arrange bacon on a slotted broiler pan or wire rack set on a rimmed baking sheet. Bake the bacon until very crisp then remove from oven and reserve. Reduce oven temp to 275°F.
Using a cheese grater, grate the potatoes onto a kitchen towel. Wring the towel over the sink to squeeze out all of the potato liquid. Put the grated potato into a bowl and add the mustard; season liberally with salt and pepper.
In a large skillet, heat about 1/8 inch of oil over medium-high heat. Once hot, spoon 4 mounds of potatoes into pan then flatten them out into hash brown cakes. Cook until brown and crisp on both sides. Transfer to a rack-lined baking sheet and keep hot in oven.
Season meat with Kosher salt and pepper, and form 4 large patties that are thinner at center for even cooking.
Heat a cast-iron skillet or griddle pan over medium-high heat. Add a little canola oil and wipe pan with paper towel to distribute it evenly. Cook patties 8 minutes, turning occasionally. Melt 2 slices of cheese on each burger; remove burgers from pan to a plate.
Heat a nonstick skillet over medium heat, add a thin pat of butter and melt. Fry eggs to medium. If using sliced tomatoes, sear them for a minute over medium-high heat after removing the eggs from the pan.
To build cheeseburgers, place cheeseburger patties on bun bottoms, then a hash brown patty, some bacon, an egg and the bun tops. (I add hoagie spread to bottom of roll and sliced tomato on top of egg.)