Meanwhile, toss shiitake and olive oil in a large bowl. Season with salt and pepper.
Light a large chimney starter full with charcoal and wait until fully ignited and coals are mostly covered in gray ash (about 20 minutes). Arrange coals under half of grill grate and place grate on top. Cover and allow to preheat for 10 minutes.
Place mushooms over hot side of grill and cook, turning frequently with tongs, until charred and tender, about 3 minutes. Transfer to cutting board and roughly chop. Place sopressata over hot side of grill and cook just until fat starts to render, about 15 seconds. Transfer to a plate.
Brush top of 1 dough round with a thin even layer of oil and season with kosher salt. Place gently on hot side of grill, oiled side down. Immediately brush top side with thin layer of oil and season with salt. Cook, checking frequently and moving to encourage even browning until bottom is dark brown and charred in spots and top is bubbly, 45 to 90 seconds. Flip using tongs or large spatula and cook until second side is charred. Flip again and transfer to cooler side of grill.
Apply 1/4 of mozzarella in a thin even layer across surface of pizza. Add 1/4 of sopressata and 1/4 of mushrooms. Cover and cook until cheese is melted, about 1 minute. Transfer pizza to cutting board with large spatula or metal pizza peel. Add 1/4 of Parmesan and 1/4 of chives. Drizzle with more oil and serve immediately. Repeat with remaining dough and toppings.