For an all-mushroom burger, make 8 portabella caps and stack several crispy mushrooms on each bun.
Dress the portabellas with salt, pepper, and lemon juice, and let stand a few minutes. Coat mushrooms in flour, then egg, and then panko mixed with parm, herbs, garlic, and red pepper flakes. Heat about 1/8 inch oil in large skillet over medium to medium-high heat and brown the mushrooms 2-3 minutes on each side, and drain on wire rack. Keep warm in 275˚F oven.
Heat a cast-iron skillet over medium-high to high heat. Form 4 loose patties of beef. Add a drizzle of oil to the skillet, add meat, season with salt and pepper, and press with a spatula; brown and flip patties, 5-6 minutes. Top with cheese to melt.