Cut the chicken breasts in half and pound out any fatter areas so the breasts are uniform in thickness.
Heat griddle until it reaches about 375 degrees F. As griddle is heating up, add a skillet to the top to warm.
Add soy sauce, mirin, granulated sugar, garlic and ginger to the warm skillet. Stir all ingredients together and cook over medium heat, stirring occasionally. When sauce begins to get foamy and bubbly (after about 5-7 minutes), move the skillet to a cooler zone on the griddle and continue to reduce until the teriyaki reaches a glaze-like consistency. Then, remove the skillet from the griddle to cool a bit. The sauce will thicken as it cools.
While sauce is cooking, add about 1 tablespoon of cooking oil to the heated griddle surface and spread it around with your spatula. Place the onion slices in the oil, sprinkle with a bit of salt, and cook the onions on the first side until a nice char develops, about 3-5 minutes. Flip to the second side and continue to cook until done. Then remove the cooked onions from the griddle. *Be careful when handling the onions so that the rings don't separate.
Place the pineapple rings on the griddle and cook on the first side until lightly golden brown and caramelized. Flip pineapple slices to the second side and cook an additional 2-3 minutes. Then remove from the griddle.
While the onion and pineapple slices are cooking, add about 1-2 tablespoons of cooking oil to a clear spot on the griddle. Place the chicken breasts in the oiled area and cook on the first side untouched for about 3-5 minutes, or until a light golden char develops. Flip the chicken breasts to the second side and continue to cook until the internal temp of each breast is 165 degrees F. *Depending on your chicken thickness and griddle temp, you may need to flip the chicken breasts a few times so they cook evenly. Then, remove the chicken from the griddle.
While chicken is still hot, immediately add the breasts to the skillet with the teriyaki sauce and toss to coat.
Build the Hawaiian Teriyaki Chicken Sandwiches. Add the teriyaki chicken breasts to the bottom toasted bun, and top with one slice of grilled pineapple and one grilled onion medallion. Add siracha if you wish, and top with the top bun. Enjoy!