In a small bowl, whisk together the olive oil, balsamic vinegar, soy sauce, 2 minced garlic cloves, smoked paprika, salt, and pepper.

Wipe the portobello caps clean with a damp paper towel. Remove the stems and scrape out the gills with a spoon so the caps sit flat and don't pool liquid.

Set aside 2 tablespoons of the marinade for basting later. Brush the rest over both sides of the mushroom caps and let them sit at room temperature for 15 minutes.

Heat a grill or 12-inch cast-iron grill pan to medium-high, about 204 C / 400 F.

Place the caps gill-side down first and grill for 4 to 5 minutes, until grill marks form and the underside releases its liquid.

Flip the caps, brush with the reserved marinade, and grill another 4 to 5 minutes until the flesh is tender all the way through and gives slightly when pressed.

If using provolone, lay a slice on each cap during the last minute and grill covered until melted.

Toast the split buns cut-side down on the grill for about 1 minute, until golden.

Stir the mayonnaise, remaining minced garlic clove, and lemon juice together in a small bowl for the aioli.

Spread aioli on both cut sides of each bun. Layer arugula, a grilled portobello cap, sliced tomato, and red onion on the bottom half.

Close the burgers and serve while the mushrooms are still warm.