Add a drizzle of extra virgin olive oil to a large non-stick saute pan and warm over medium-high heat. Add the sliced mushrooms, season with a sprinkle of kosher salt and pepper, and cook for about five minutes, stirring occasionally, until golden brown. Remove the mushrooms from the skillet and set aside on a plate.
Add 2 more tablespoons of olive oil to the skillet and add the onion. Reduce the heat to medium and saute until the onions are soft and just turning golden, about 5 minutes. Add the carrots and celery and saute another 5 minutes until softened.
Increase the heat to medium-high and add the bell pepper. Cook for 2-3 minutes until the bell pepper just starts to soften.
Reduce the heat back to medium and add the garlic and swiss chard. Cook, stirring to incorporate the greens, for about 1 minute or until the chard has started to wilt. Add the garbanzo beans, olives, green onions, parsley, cherry tomatoes, and reserved mushrooms. Let cook for 1 minute until warmed.
Turn off the heat, drizzle the juice of 1 lemon on top, stir in the parmesan and mozzarella cheeses, and season well with kosher salt and ground black pepper. Enjoy right away with a hunk of crusty french bread on the side.