Heat a large skillet over medium heat and add butter and olive oil. Add the squash, shallots, garlic, pepper flakes, salt and pepper and stir to coat. Cook until the squash is just tender, stirring occasionally, about 10 minutes. Stir in a few grates of fresh nutmeg.
Transfer the entire pan contents to a food processor. Pulse until just slightly chopped (the squash will start to somewhat puree). Add in the beans and pulse just a few times until they are coarsely chopped. Transfer the mixture to a bowl and stir in the egg, bread crumbs and flour until your have a slider “dough.” Form the dough into 8 to 10 sliders (I go by the size of buns I have) and place them on a baking sheet lined with parchment paper. Stick them in the fridge for 30 minutes.
After 30 minutes, heat the same skillet over medium heat. Add another drop of olive oil if needed, then carefully add the sliders into the skillet. Cook until golden on each side – about 6 to 8 minutes per side – gently flipping with a small square spatula so you don’t bread the sliders apart. In the last minute or so of cooking, add sliced havarti cheese on top to melt.
To serve the sliders, toast the buns slightly. Spread the ketchup on the bottom bun then top with a burger. Add the cilantro on top of the cheese (or lettuce if you hate cilantro!) and eat.