Add the butter to a sauté pan with the sliced white onions, white granulated sugar, and Kosher salt.
*The white granulated sugar helps the onions caramelize faster.
Reduce the heat to medium, and cook the onions for roughly thirty minutes, stirring occasionally.
*If the onions are getting inconsistently dark, add a splash or two of water and stir to combine. Avoid burning the onions at this stage.
While the onions are cooking, boil one whole Yukon gold potato in a pot of salted water.
When the onions are golden-brown in color and caramelized, remove them from the heat.
When the potato is fully cooked through in the center, turn off the heat under the boiling water and keep the potato sitting in the hot water while you prepare the remaining patty ingredients.
Boil the dried textured soy protein in enough water to cover by one inch.
Cook the textured soy protein for roughly ten minutes, or until it is completely soft.
When the textured soy protein is cooked, drain it from the liquid.
Finely dice the cooked potato and textured soy protein, and add the diced potato and textured soy protein to a large mixing bowl.
Add the caramelized onions from earlier.
Add the whole egg, one Tablespoon of butter, one cup of shredded cheddar cheese, along with some red pepper flakes, dried oregano, granulated garlic, ground cumin, ground cayenne, and sliced chives.
Mix the contents well, until everything is evenly distributed.
Taste, and add more Kosher salt if desired.
When you are satisfied with the taste of your patty mixture, form it into patties on a parchment paper-lined sheet tray, and stick the sheet tray with the shaped patties into the fridge to firm up.
When you are ready to prepare you breakfast sandwiches, butter the sesame seed buns.
Griddle the buns until they are golden-brown.
Pan-fry the prepared breakfast sandwich patties you made earlier in some butter until they are crispy and lightly griddled on the outsides.
Assemble the sandwiches with your toasted buns, meatless patties, smoked Gouda cheese, sunny-side-up egg, lettuce, and tomato.
Serve with ketchup and hot sauce, if desired.