Heat a tablespoon of olive oil in a skillet over medium heat. Add the sliced mushrooms and sauté until golden brown, about 5-7 minutes. Stir in the grated carrots and red onion, cooking for another 2-3 minutes. Season with salt and pepper to taste. Remove from heat and let cool slightly.
In a food processor, combine the drained cannellini beans and dry-roasted cashews. Pulse until you achieve a chunky paste.
In a large mixing bowl, combine the sautéed vegetables, the bean mixture, cooked brown rice, panko breadcrumbs, grated Parmesan cheese, egg, mayo, garlic powder, and paprika. Stir until well mixed.
Using your hands, form the mixture into burger patties, about 1/2 inch thick. You should get around 6-8 patties.
Preheat your oven to 375°F (190°C). Place the patties on a baking sheet lined with parchment paper. Bake for 20-25 minutes, flipping halfway through, until golden brown and firm.
Serve your veggie burgers on whole grain buns, lettuce wraps, or on their own. Top with your favorite condiments and enjoy!