Make the mayo:
In a small bowl, whisk the mayo, sriracha, lemon juice, mustard, and garlic. Chill until ready to use.
Heat 2 tablespoons of the oil in a large skillet over medium heat. Add the onions and ¼ teaspoon each salt and pepper cook, stirring, until deeply golden and lightly caramelized, 11-12 minutes. Place the onions in a large bowl, cool for 5 minutes, then remove 1/2 cup of the onions to a small bowl but leave the rest in the large bowl. Add the remaining 2 tablespoons of oil to the skillet. Add the kale, mushrooms, scallions, and remaining ¼ teaspoon each salt and pepper to the skillet and cook, stirring, until the kale is wilted and the mushrooms have softened and released their water, 6-7 minutes Add the garlic and cook, stirring, until softened, 1-2 minutes. Transfer the kale mixture to the large bowl with the onions.
Add the mashed potatoes, carrots, and cheese, season with salt and pepper, and, using lightly oiled hands, form into six equal-sized patties. Place flour in one bowl and season with salt and pepper. Place the panko bread crumbs in another bowl and season with salt and pepper. Place the egg in a third bowl. Dip the patties in the flour, shaking off the excess, then dip in the egg. Letting the excess drip off. Press the patties firmly into the panko mixture on both sides.
Fry the patties and make the sandwiches:
Heat ¼ inch oil in a large skillet over medium heat. Gently fry the breaded patties until golden, 2-3 minutes per side; drain on a towel-lined plate. Strange about 1 ½ tablespoons of the reserved onions on the bottom side of each bun, then layer with 2 lettuce leaves and a fried patty. Spread the top of each bun with mayo and close the sandwiches.