Preheat the oven to 475°F. Wash and dry the fresh produce. Halve the buns. Halve the potato lengthwise; cut crosswise into ¼-inchthick pieces. Cut the broccoli into small pieces. Drain and rinse the beans; transfer to a large bowl and mash with a fork to your desired consistency. Grate the cheese (discarding any rind). Finely chop the peppers. Peel the garlic; mince 3 cloves. Mince the remaining clove; smash with the side of your knife until it resembles a paste (or use a zester). In a bowl, combine the mayonnaise, peppers and half the garlic paste; season with salt and pepper to taste.
Line a sheet pan with aluminum foil. Place the potato on the foil-lined sheet pan. Drizzle with olive oil and season with salt and pepper; toss to thoroughly coat. Arrange in a single, even layer and roast 24 to 26 minutes, or until browned and tender when pierced with a fork. Remove from the oven, leaving the oven on.
In a large pan (nonstick, if you have one), heat 1 teaspoon of olive oil on medium-high until hot. Add the broccoli; season with salt and pepper. Cook, stirring occasionally, 4 to 6 minutes, or until lightly browned. Add the beans, minced garlic and as much of the red pepper flakes as you’d like, depending on how spicy you’d like the dish to be; season with salt and pepper. Cook, stirring occasionally, 2 to 4 minutes, or until the beans begin to brown. Add ¾ cup of water; cook, stirring occasionally, 4 to 6 minutes, or until the water has cooked off. Transfer to a bowl. Wipe out the pan.
Add the egg and breadcrumbs to the bowl of cooked broccoli and beans; season with salt and pepper. Stir to thoroughly combine. Using your hands, gently form the mixture into two 1-inch-thick patties. Transfer to a plate. In the same pan, heat 1 teaspoon of olive oil on medium-high until hot. Add the patties and cook 3 to 4 minutes per side, or until browned and cooked through. Leaving any browned bits (or fond) in the pan, transfer to a plate. While the patties cook, in a bowl, combine the cheese and remaining garlic paste; toss to combine. Carefully top the roasted potato with the cheese mixture. Return to the oven and bake 3 to 5 minutes, or until the cheese is melted and lightly browned. Remove from the oven and divide between 2 dishes.
Heat the pan of reserved fond on medium-high until hot. (If the pan seems dry, add 1 teaspoon of olive oil.) Add the buns, cut sides down, and toast 1 to 2 minutes, or until golden brown. Transfer to a work surface. Top the bun bottoms with the aioli, cooked patties and bun tops. Divide between the dishes of finished potato. Enjoy!