Prepare the Brussels Sprouts: Preheat the oven to 450°F (230°C) with a rack in the center position. Place the halved Brussels sprouts on a large baking sheet. Drizzle them with 2 tablespoons of toasted sesame oil and toss with your hands to distribute the oil evenly. Season lightly with salt and freshly ground pepper. Flip the Brussels sprout halves so the the cut side is facing down, this will help improve caramelization.
Roast the sprouts for 15 to 25 minutes, flipping them halfway. Roasting time will vary depending on the size of the sprouts. Place on a rack and allow the sprouts to cool slightly while you prepare the dressing.
Prepare the Dressing: In a small bowl, whisk together the soy sauce, rice vinegar, fresh ginger, miso paste, and pepper flakes. Slowly pour in remaining 2 tablespoons of sesame oil, whisking continuously with your other hand, until the ingredients are emulsified. Season with pepper to taste.
Prepare the Shiitake Mushroom Topping: Remove the stems from the mushrooms and using a sharp chef’s knife, slice the mushrooms lengthwise into paper-thin strips. Line a small baking sheet with paper towels and set aside.
Heat oil in a small to medium saucepan - it should reach halfway up the sides - over medium heat until the temperature reaches 350°F (175°C).
Carefully fry a handful of mushroom slices at a time (be careful, as the oil can spit). The oil will bubble very strongly at first and start to become more quiet as mushrooms pieces are nearly done frying. Using a slotted stainless steel spoon or spider skimmer, gently flip the mushrooms continuously in the oil until they begin to curl slightly and turn golden brown.
Scoop the mushroom pieces from the oil and transfer them to the paper-towel lined baking sheet to drain the excess oil. Immediately season with salt. Fry the remaining sliced mushrooms in batches as needed.
Gently place the still warm roasted Brussels sprouts in a large bowl or serving platter. Pour on the sauce and toss gently to coat. Season to taste and garnish with the fried shiitake mushrooms. Serve immediately.