1. Slice mushroom caps in half and place in a wide-rimmed bowl. Combine oil, paprika, garlic powder, cumin, and black pepper in a small bowl. Brush mushrooms with oil-spice mixture, and let sit 15 minutes.
2. Next, heat a greased grill or grill pan over medium-high heat. Add mushrooms (in batches, if necessary, to avoid over-crowding). Cook 3 to 4 minutes, undisturbed, until grill marks appear, flip and cook 2 more minutes. Season with ½ tsp. of the salt, transfer to a cutting board, and slice into strips.
3. Mash avocados in a bowl with ¼ tsp. salt, lime zest, and juice; set aside.
4. To assemble tacos, spread avocado evenly on tortillas and top with mushrooms and red cabbage. Garnish with radishes and jalapeños, if desired.