Fire up a gass or charcoal grill or indoor grill pan.

Brush the mushroom caps with olive oil and season with salt. Grill over medium-high heat until soft and nicely browned, about 5 to 8 minutes. Set aside under foil.

Brush the onion slices with the oil and season with salt. Grill over medium-high heat until charred and soft, about 3 minutes. Set aside under foil.

Grill the burger buns until toasty, about 2 minutes.

When ready to serve, return the mushrooms to the grill over medium-low heat and top with the cheese slices. Cover with the hood so the cheese can melt. Alternatively, if cooking indoors, place the mushrooms on a baking sheet and melt cheese under the broiler for 1 minutes.

Assemble the burgers: Slather the romesco sauce on both buns, top the bottom half with the mushroom cap, followed by the arugula. Sandwich together and serve with the remaining sauce on the side.