Preheat the oven to 425 degrees F. Toss the eggplant with the oil, salt and pepper the roast until softened, 25 - 40 minutes.
Add the roasted eggplant to the bowl of a food processor fit with the steel blade and pulse a few times. Add the shallot and white beans and process until almost smooth. Add the remaining ingredients and process until a dough forms (this will be a little soft, but that's okay).
Using wet hands, shape the dough into patties and place them on a parchment lined baking sheet. Chill in the refrigerator for at least 1 hour (this will help them firm up).
15 minutes prior to cooking the burgers, preheat the oven to 400 degrees F. Remove the baking sheet and bake in the center of the oven for 30 minutes, flipping the patties halfway through.