Add 3/4 of the beans to a food processor and pulse until texture is almost like a paste. Does not have to be completely smooth. Transfer to a large mixing bowl.
Add the breadcrumbs, onion, cumin, chili, salt, and pepper to the bowl and mix well until completely incorporated. Fold in the remaining beans.
Wet hands and shape bean mixture into 4 patties of equal size. Set aside.
Heat a large skillet over medium heat and add 2 tablespoons of oil. Add garlic and sauté for 1 minute. Add the spinach and sauté until wilted, about 4 minutes. Remove from heat.
Heat a separate skillet over medium-high heat. Let the pan heat until searing hot, then add remaining 2 tablespoons oil. Add the patties and brown and cook about 3 to 4 minutes on each side. Transfer to paper towels to allow excess oil to drain.
Place one patty on each of the bottoms of the buns. Top with a portion of the sautéed spinach. Sprinkle Parmesan over each and top with buns. Serve immediately.