Heat a 10-inch nonstick skillet, with oven safe handle over medium to medium-high heat. Wrapping a rubber handle in foil will allow the pan to transfer to oven as well.
To hot pan, add oil and mushrooms. Season mushrooms with thyme, salt and pepper. Cook 3 to 5 minutes until all of the veggies are tender. Beat eggs with salt and pepper, dash of half-and-half, lemon zest, scallions and hot sauce. Pour over the mushrooms. Lift and settle eggs in the pan as they brown on the bottom. When the eggs are set but remain uncooked on top, transfer to oven for 7 or 8 minutes, until frittata is golden brown and puffy. Sprinkle the frittata with cheese and return to the oven for 1 or 2 minutes.
Serve frittata from the hot pan and cut into 6 wedges at the table.