Combine all ingredients for the teriyaki sauce in a small saucepan, except for the cornstarch. Bring to a gentle boil over medium heat, then reduce to a simmer over low heat.

Whisk together cornstarch and water and add to saucepan. Whisk until thickened, then remove from heat.

In a large mixing bowl, combine all ingredients for the burger patties and mix until combined. Start with 1/3 cup breadcrumbs. Mixture should be thick enough to shape into patties without much of it sticking to your hands. If it's very sticky, add in additional breadcrumbs a tablespoon at a time until desired texture is reached.

Shape mixture into four patties about 3/4" inch thick. Refrigerate for at least 20 minutes. This will allow the patties to firm up as the breadcrumbs absorb the liquid and bind everything together.

In the meantime, mix together the ingredients for the teriyaki mayo and place in the refrigerator until time to serve.

When it comes time to grill, brush sliced onions and pineapple with olive oil. Place the onions on first, as they take the longest to cook. (They will require 8-10 minutes per side, until slightly charred.)

Brush patties with teriyaki sauce and grill 3-5 minutes per side, until there are grill marks. Baste the patties with more teriyaki sauce after flipping. If you're adding vegan cheese, do so during the last 1-2 minutes of grilling to give it time to melt.

You'll also grill your pineapple slices with the burger patties, until there are grill marks.

To assemble burgers, spread the buns with teriyaki mayo. Each burger gets one patty, one to two slices of grilled pineapple, and one or two thick slices of grilled onion. Add additional toppings as desired and enjoy.