Heat the olive oil over medium-high heat in a large skillet. Add in the onion, carrots, and garlic and cook until tender and fragrant, about five minutes. Add in the mushrooms and cook until they release almost all of their water, about 10 minutes. Remove from heat and set aside.
In the basin of a food processor, combine the wild rice and three cups of the black beans. Pulse until it resembles a chunky paste. In a large bowl, mix together the mushroom mixture, the rice mixture, the remaining black beans, and all remaining ingredients until well combined. Cover and refrigerate for a half hour.
Preheat oven to 350°F. Line a baking sheet with parchment paper, set aside.
Form the chilled mixtured into 10 even patties. Place on the prepared baking sheet, and bake in preheated oven for 20-25 minutes, flipping halfway in between, or until the outsides are just barely golden brown.