Heat broiler. Line a baking pan with foil; coat foil with nonstick cooking spray. Wrap rolls in foil and place in lower part of oven to warm.
In a mini chopper, combine mushrooms, 2 of the scallions and the ginger. Pulse until finely chopped. Transfer to a medium-size bowl and add pork, 1/4 cup of the stir-fry sauce and the bread crumbs. Stir to blend.
Shape mixture into 12 equal-size patties, a scant 1/4 cup for each. Place patties on prepared baking pan; broil 4 inches from heat for 6 minutes, turning once, until instant-read thermometer registers 160° when inserted into center of burgers.
Meanwhile, stir together mayonnaise, remaining 1 teaspoon teriyaki stir-fry sauce and the sugar in a small bowl. Split rolls and top each with a burger. Spread 1 teaspoon teriyaki mayonnaise on each roll; sprinkle with scallion.
