Place mushroom caps in a 9x13-inch baking dish. In a medium bowl, combine olive oil, balsamic vinegar, Italian seasoning, garlic powder, salt, and pepper; mix to combine. Pour marinade over mushrooms and use a pastry brush to brush on both sides, gently flipping mushrooms each time, and ending in stem-side-up position. Let sit, uncovered, at room temperature for at least 30 minutes (or up to 2 hours).

Combine all romesco sauce ingredients in a blender or food processor. Blend until mostly smooth (a few chunks are fine). Set aside.

Heat olive oil in a large skillet over medium heat. Once hot, add sliced onion and a pinch of salt. Sauté, stirring occasionally, until soft and slightly crisp around the edges, about 8 to 10 minutes. Transfer to a bowl.

Return the skillet to medium heat. Once hot, add portobello mushrooms and sauté for about 4 to 5 minutes on each side, or until the mushrooms are deep golden brown on the edges and softened/reduced in size. While cooking, brush with any leftover marinade. Top each mushroom with a slice of cheese and place lid on pan. Cook for 1 additional minute, or until cheese is melted.

Lightly toast each half of bun. Spoon a generous portion of romesco sauce on the bottom bun and a swipe of hummus or mayonnaise on top bun. Stack bottom bun with one portobello mushroom, caramelized onions, and arugula or microgreens. Finish with top bun and enjoy immediately.