In a covered microwave-safe bowl, combine the 1 cup chicken broth and 1/2 ounce dried shiitake mushrooms and microwave until steaming, 1 to 2 minutes. Let sit until softened, 5 to 10 minutes. Drain the mushrooms in a fine-mesh strainer and reserve liquid.
In a food processor, pulse the 2 slices hearty white sandwich bread until finely ground, 10 to 15 pulses. Dump the breadcrumbs into a large bowl (don't wash the processor bowl). Add the 1 ounce Parmesan cheese, 1 tablespoon chopped fresh Italian parsley, 1 1/2 teaspoons unflavored gelatin*, 1 teaspoon salt, and 1/4 teaspoon pepper to bowl with breadcrumbs and mix until thoroughly combined.
In the food processor, combine the mushrooms and half of the anchovies and pulse until chopped fine, 15 to 20 pulses.
Add the mushroom mixture, 1 1/2 pounds ground turkey, 1 large egg, and half of the minced garlic to the bowl with the breadcrumb mixture and mix with your hands until thoroughly combined.
Divide mixture into 16 portions (each about 1/4 cup loosely packed). Using your hands, roll each portion into a ball, taking care to not pack them too tightly. Put the meatballs on a large plate and refrigerate for 15 minutes.
In the food processor, pulse the one (14.5-ounce) can whole peeled tomatoes and their juice to a coarse puree, 10 to 15 pulses.
In a small bowl, combine the 1/2 teaspoon dried oregano, 1/8 teaspoon red pepper flakes, the remaining anchovies, the remaining garlic, and 1/4 teaspoon black pepper.
In a 12-inch (30-cm) nonstick skillet over medium-high heat, warm the 3 tablespoons extra-virgin olive oil until shimmering, 2 to 4 minutes. Add the meatballs and cook until well browned all over, 5 to 7 minutes. Use a slotted spoon to transfer meatballs to a paper towel-lined baking sheet, leaving fat in the skillet.
☞ TESTER TIP: If your turkey is very lean, you may need to add a little extra oil to the skillet.
Add oregano mixture to the skillet and cook, stirring constantly, until fragrant, about 30 seconds. Increase heat to high, stir in the 2 tablespoons tomato paste, reserved mushroom liquid, and pureed tomatoes, and bring to a simmer.
Return meatballs to the skillet, reduce heat to medium-low, cover, and cook until the meatballs register 160°F (71°C) on an instant-read thermometer, 12 to 15 minutes, turning meatballs once.
Place meatballs on a serving platter, increase heat to high, and simmer sauce until slightly thickened, 2 to 5 minutes. Stir in the 1/4 cup chopped fresh basil and season with granulated sugar, salt, and pepper to taste. Pour sauce over meatballs and serve, passing extra Parmesan separately.
Notes
*Why would you add gelatin to Italian-style turkey meatballs?
Mainly, adding gelatin to ground meat helps it to retain moisture resulting in a more tender texture. Adding a small amount of unflavored gelatin mitigates graininess by trapping moisture and giving the meatballs a juicy mouthfeel. Then, a 15-minute refrigeration gives it all time to bind together and produce a springy, juicy texture in the cooked meatballs.