In a blender, puree the onion, garlic, tamari, mirin, 1 tablespoon gochujang paste, brown sugar, and ginger. Season with salt and pepper. Transfer to a bowl and set aside.
Soak 4 to 6 bamboo skewers in water for 30 minutes. Skewer the trumpet mushrooms, dividing them between the bamboo skewers (this makes them easier to grill). Brush the mushrooms all over lightly with the marinade. (The mushrooms can sit at room temperature for up to 1 hour.)
To make the sauce, transfer the remaining marinade (you should have about ⅓ cup) to a small saucepan and add the hoisin, miso, vinegar, and sesame oil. Bring to a simmer over medium-low heat, whisking to combine. Cook, stirring, until the flavors combine, about 3 minutes. Taste and adjust the seasoning with more gochujang, vinegar, sugar, and/or sesame oil. If the sauce is too thick to drizzle, loosen it with a little water. Set aside.
Light a Quick Briq or Everyday Briq and set the temperature to 550°F. Brush the grill grate with oil. Drizzle the mushrooms lightly all over with oil. Cook the mushrooms, turning once, until grill-marked all over and tender, 7 to 9 minutes (depending on the thickness).
Arrange the skewers on a platter, drizzle with some of the sauce, and serve. Alternatively, remove the mushrooms from the skewers, slice thinly lengthwise, and use them on pizza or flatbread, drizzled with the sauce.