For the patties: Lightly oil 1 quarter-size sheet pan and keep in a warm place.
In a heavy-bottomed pot over high heat, mix the chickpea flour, salt, cumin, black pepper and cayenne. Gradually add 2 1/2 cups water while stirring with a flat-bottomed wooden spatula. Avoid lots of lumps of flour, but a few are ok. While stirring almost constantly, gradually lower the heat as the mixture thickens. Add the parsley when most of the water has been absorbed. Once it is thickened and the mixture pulls away from the sides and bottom of the pot when you stir it, it is done. It should take 10 to 15 minutes.
Transfer the patty mixture to the sheet pan and spread it evenly, covering the entire pan. (Optional: Also oil the bottom of another pan and place it on top of the patty mixture. Press firmly to create a smooth, even surface, then remove.) Cool completely.
For the squash: Preheat the oven to 350 degrees F.
Cut the squash in half lengthwise and remove the seeds. Cut crosswise into 1/4-inch slices (when delicata first come into season you can eat the skin). Toss with a little olive oil and a pinch of black pepper, place on a sheet pan and bake until easily pierced with a knife, 10 to 15 minutes. Keep these warm.
For the aioli: Add the mustard, garlic, egg yolks, lemon zest and lemon juice to a food processor and puree. With the unit running, slowly drizzle in the olive oil until all is used. It should form a thick emulsion, like mayonnaise. Cool completely.
For the slaw: Toss the carrots and radishes with the oil and vinegar. Season with salt and pepper. Keep at room temperature.
Now for the sandwiches! Fill a deep-fat fryer with vegetable oil (alternatively use rice bran oil) and set to 350 degrees F. (If preferable, bake or grill the patties as you would polenta.)
Slice the rolls in half to open them, lightly brush with olive oil and toast them either on a grill or in the broiler of an oven. Take the sheet pan of chickpea and cut into 12 pieces. For each sandwich, drop 2 of the patties into the fryer until they are crispy and begin to brown, about 4 minutes. Spread aioli on each side of the roll, then place salad greens on each side. Cover 1 side with the carrot radish slaw, then place 3 to 4 slices of roasted squash on top. Remove the patties from the fryer, drain on paper towels and lightly salt them. Place them over the squash and close the sandwich up!
This recipe was provided by professional chefs and has been scaled down from a bulk recipe provided by a restaurant. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.