Place turkey pieces on large plate in single layer. Freeze meat until very firm and hardened around edges, 35 to 45 minutes. Meanwhile, sprinkle gelatin over chicken broth in small bowl and let sit until gelatin softens, about 5 minutes. Pulse mushrooms in food processor until coarsely chopped, about 7 pulses, stopping and redistributing mushrooms around bowl as needed to ensure even grinding. Set mushrooms aside; do not wash food processor.
Pulse one-third of turkey in food processor until coarsely chopped into 1/8-inch pieces, 18 to 22 pulses, stopping and redistributing turkey around bowl as needed to ensure even grinding. Transfer meat to large bowl and repeat two more times with remaining turkey.
Return 1/2 cup (about 3 ounces) ground turkey to bowl of food processor along with softened gelatin, soy sauce, and baking soda. Process until smooth, about 2 minutes, scraping down bowl as needed. With processor running, slowly drizzle in oil, about 10 seconds; leave paste in food processor. Return mushrooms to food processor with paste and pulse to combine, 3 to 5 pulses, stopping and redistributing mixture as needed to ensure even mixing. Transfer mushroom mixture to bowl with ground turkey and use hands to evenly combine.
With lightly greased hands, divide meat mixture into 6 balls. Flatten into 3/4-inch-thick patties about 4 inches in diameter; press shallow indentation into center of each burger to ensure even cooking. (Shaped patties can be frozen for up to 1 month. Frozen patties can be cooked straight from freezer.).
FOR A CHARCOAL GRILL: Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.
FOR A GAS GRILL: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s).
Clean and oil cooking grate. Brush 1 side of patties with oil and season with salt and pepper. Using spatula, flip patties, brush with oil, and season second side. Place burgers over hot part of grill and cook until burgers are well browned on both sides and register 160 degrees, 4 to 7 minutes per side. (If cooking frozen burgers: After burgers are browned on both sides, transfer to cool side of grill, cover, and continue to cook until burgers register 160 degrees.).
Transfer burgers to plate and let rest for 5 minutes. While burgers rest, grill buns over hot side of grill. Transfer burgers to buns, add desired toppings, and serve.