Roughly chop the red onion and core the bell pepper; trim and roughly chop the mushrooms.
Place the chopped onion, bell pepper, mushrooms, rinsed black beans, garam masala, and salt in a food processor and pulse until the mixture is finely minced but not pureed.
Transfer the minced vegetable and bean mixture to a large bowl and fold in the silken tofu and panko until combined; taste and adjust garam masala or salt if desired.
Refrigerate the mixture for at least 20 minutes to firm up; if too wet after chilling, stir in up to 1/4 cup additional panko.
Shape the chilled mixture into 6 to 8 patties, depending on desired size.
Heat a frying pan over medium heat and add the cooking oil, swirling to coat the pan.
Fry the patties in batches without crowding, cooking 2–3 minutes per side until golden brown and heated through.
Assemble each bun with mayonnaise, a patty, arugula, and goat or soy cheese, then serve.
