Place the bulgur in a medium bowl, cover with boiling water, and let sit until the bulgur has absorbed the liquid and fluffed up. Depending on the fineness/coarseness of the grind, this can take anywhere from 15 minutes to 1 hour. It should yield about 1⁄2 cup of hydrated bulgur.
Heat the olive oil in a large skillet over medium-high heat. Add the mushrooms and onions, season with the soy sauce and Worcestershire sauce, and cook until the mushrooms have wilted and most of the liquid has cooked off, about 7 minutes. Stir in the garlic and cook for another 30 seconds. Transfer the mushroom mixture to a bowl and set aside.
Drain the bulgur thoroughly and add it to the mushrooms. Mix in the Parmigiano-Reggiano cheese, breadcrumbs, Italian seasoning, parsley, eggs, and a few grinds of black pepper. Cover the bowl and place in the refrigerator to rest for at least 4 hours or overnight.
Cook the meatballs:
Preheat the oven to 400°F with a rack placed in the center of the oven. Line a sheet pan with parchment paper.
Using a small ice cream scoop or large spoon, scoop 2-tablespoon portions of the mixture onto the prepared pan. Using your hands, roll each portion into a ball, pressing to compact it.
Bake for 15 minutes, or until the meatballs are firm to the touch.
Bring the marinara sauce to a bubble in a large saucepan over medium heat. Transfer the meatballs to the sauce and lower the heat to a simmer. Cover and let the meatballs simmer in the sauce for 45 minutes.
Garnish the meatballs with chopped parsley and serve with toasted bread.