In a large jar, combine the sugar, salt, and hot water; stir to dissolve. Add the vinegar and onion. Let sit while you prepare the other ingredients, at least 30 minutes. Or chill for up to 1 week.
Combine the cashews and 1/3 cup water in the bowl of a high-speed blender and puree until very smooth, adding more water if needed. Chill if the mixture is warm. Add the cilantro, juice of 1 lime, and salt, and blend until very smooth. Taste, adding more salt or lime if you like. Add enough water to make the crema thin enough to drizzle. Chill until needed, up to 1 week.
Brush the sliced mushrooms with oil on both sides. Sprinkle with salt if using.
Preheat your grill to medium (or heat a grill pan over medium heat). Lay the mushrooms on the grill and cook until browned on the bottom with grill marks, 6-10 minutes, adjusting the heat as needed. Flip and cook on the second side until the mushrooms are tender and charred on both sides, another 5 minutes or so.
Remove the cooked mushrooms and while hot, brush all over with the BBQ sauce. Cover and keep warm.
Char the tortillas over a gas flame or in a hot skillet and stack in a clean kitchen towel to keep them pliable and warm.
To assemble the tacos, fill the tortillas with BBQ mushrooms, avocado, pickled onion, avocado, crema, radish, and cotija. Serve with lime wedges for squeezing.