Pulse the edamame in a food processor until it's coarsely chopped. Transfer the edamame to to a large bowl. Stir in the rice, sesame seeds, soy sauce, garlic, and ginger. Fold in the egg.
Line a large baking sheet with parchment paper. Divide the burger mixture onto the baking sheet in 4–6 mounds (4 for bigger burgers, 6 for smaller ones), then press each mound into a patty. Bake the burgers for 15 minutes, then brush with the teriyaki marinade and bake for 10 minutes more.
While the burgers are baking, heat an indoor grill to Sear (or High if yours doesn't have a Sear setting; a grill pan works too if you don't have an indoor grill). Spray the grates with oil and grill both sides of the pineapple and onions until the pineapple is has grill marks and the onions are crispy. Break the onions into small pieces.
Toss the coleslaw mix with the vinegar, sesame oil, and salt and pepper to taste.
Place each burger patty on a bun, then top with grilled pineapple, slaw, and crispy green onions.