Chop the vegetables as directed.
Melt the butter in a large frying pan over medium-high heat. Add the onion, carrot and pepper. Saute 1-2 minutes. Add the garlic, and saute another 1-2 minutes. Add the tarragon and paprika. Lower the heat and let the vegetables cook slowly and soften. Add more butter if necessary.
Meanwhile, mix the eggs, water, soy sauce and salt and pepper in a bowl.
Add the mushrooms into the vegetables in the frying pan. Cook another 5-10 minutes on low heat. Cover pan to retain moisture.
Spread vegetables in pan with spatula so they are pressed down and as flat as possible. Pour the egg mixture over them. Sprinkle the cheese over the top.
Turn the heat up slightly. Using a metal spatula, gently lift the edges of the omlete that is forming. Tilt the pan to let the unset egg flow down against the edges.
Cover the pan and let the egg set further. This should take another 5 minutes or so.
Lift around the edges with the spatula to loosen the omelet. Slide omlet on to a serving plate. Serve with crusty bread, and a nice white wine (if you partake).