Preheat a grill pan or skillet over medium-high heat (about 400°F / 205°C).
Gently wipe the portobello mushroom clean and remove the gills if desired.
Brush both sides of the mushroom with olive oil and season with salt, pepper, and garlic powder.
Place the mushroom gill-side down and cook for 4–5 minutes until browned with grill marks.
Flip the mushroom and cook another 4–5 minutes, draining excess liquid if needed.
Add the provolone slice on top during the last 1–2 minutes; cover to melt the cheese.
Toast the brioche bun cut-side down for 30–60 seconds until lightly golden.
Assemble the burger: bottom bun, cheesy portobello, roasted red peppers, arugula, and balsamic glaze.
Top with the remaining bun and serve immediately while hot.