Arrange the red peppers on a foil-lined baking sheet. Broil on high, flipping occasionally, until the peppers are blackened and puffy, about 5-10 minutes.
Remove the peppers from the oven and use tongs to peel off the blackened skin. If cooked through, the skin should peel off easily. If not, continue to broil. Cut off the tops and squeeze out the seeds.
Place the peppers into a food processor with the mayonnaise, whole grain mustard, celery, capers, pickle slices, and minced garlic. Season with salt and pepper to taste. Blend until mostly smooth, leaving chunks as desired. Chill until ready to serve.
Pour one tablespoon of oil in a hot skillet over medium heat. Add the chopped onions and cook until browned and caramelized, stirring occasionally, about 15 minutes.
In a small bowl, whisk together the three regular eggs. Have flour seasoned with salt and pepper in a separate bowl. In another bowl, stir together the panko and Cajun seasonings. Pat the alligator sirloin dry, then dip into the flour until covered, then the eggs, letting the excess drip off, then place into the panko. Flip until completely covered. Let rest in the panko until ready to cook.
Preheat the oil over medium-high heat in a large skillet. Place the breaded sirloins in the hot oil. Cook on each side until golden brown, about 4-5 minutes per side, until cooked to 160-165°F.
While the alligator cooks, toast the slider buns. Assemble the cooked alligator on the slider buns with shredded cabbage, remoulade, caramelized onions, and pickles, as desired. Use a toothpick to help keep it all together. Serve hot - enjoy!