Preheat grill to medium-high heat. Lightly brush with oil.
In a small bowl, whisk together seasoning, vinegar, and oil. Sauce will be thick.
Lay mushroom caps gill side down. Brush or spread half the sauce on the mushroom caps. Try to get as much sauce as you can in gills. Head to the grill!
Place mushroom caps on grill, gill side up. Grill caps, with lid closed, 5–6 min. Flip caps and grill 5 more min.
Flip caps over (gills should be facing up). Add 1 slice Swiss cheese per cap and grill until melted, 1 min. Caps should be tender.
Tuck mushroom caps into buns and add toppings, if desired.
