Cook for about 3 minutes until mushroom softens and season with salt and pepper and basil.
Crack eggs into a bowl and add a splash of milk. Mix well.
Heat a 10 inch non-stick skillet over medium flame. Add 2 pats of butter. Add the eggs and spread by tipping the pan until the bottom is evenly covered. Let the eggs set over medium heat. After 30 seconds or so, start shaking the pan occasionally so the eggs slide around and don't stick.
When the eggs are set (don't brown the omelet), add the filling in a line down the middle of the pan.
Use a plastic spatula to fold a third of the omelet over the filling and remove the pan.
Tip the pan until the omelet slides onto a plate and use the edge of the pan to fold the last 1/3 of the omelet over the filling, until the omelet looks like a tube on the plate.
Season lightly with salt and pepper and garnish with basil.