Place turkey pieces on a large plate in a single layer.

Freeze meat until very firm and hardened around the edges, about 35-45 minutes.

Sprinkle gelatin over chicken broth in a small bowl and let sit until gelatin softens, about 5 minutes.

Pulse mushrooms in a food processor until coarsely chopped, about 7 pulses, stopping and redistributing mushrooms as needed.

Set mushrooms aside.

Pulse one-third of the turkey in the food processor until coarsely chopped into 1/8-inch pieces, about 18-22 pulses, stopping and redistributing turkey as needed.

Transfer meat to a large bowl and repeat two more times with remaining turkey.

Return 1/2 cup (about 3 ounces) ground turkey to the bowl of the food processor along with softened gelatin, soy sauce, and baking soda.

Process until smooth, about 2 minutes, scraping down the bowl as needed.

With processor running, slowly drizzle in oil, about 10 seconds.

Return mushrooms to food processor with paste and pulse to combine, 3 to 5 pulses, stopping and redistributing mixture as needed.

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 a shallow indentation into center of each burger.

Open bottom vent completely for charcoal grill.

Light a large chimney starter filled with charcoal briquettes (6 quarts).

When top coals are partially covered with ash, pour evenly over half of the grill.

Set cooking grate in place, cover, and open lid vent completely.

Heat grill until hot, about 5 minutes for charcoal grill.

Turn all burners to high, cover, and heat grill until hot, about 15 minutes for gas grill.

Leave primary burner on high and turn off other burner(s) for gas grill.

Clean and oil cooking grate.

Brush one side of patties with oil and season with salt and pepper.

Flip patties, brush with oil, and season the 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.

Transfer burgers to a 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.