Combine oil, mushrooms, thyme, garlic, salt, and pepper in a bowl. Roast in preheated oven until deeply caramelized, about 40-45 minutes. Stir occasionally.
In a bowl, whisk the mayonnaise, Dijon mustard, shichimi togarashi, and honey.
Cook the chicken burger per specifications.
Divide mayo mixture on the top and bottom of the onion roll. Place burger on roll and top with mushroom mixture.