In a 8" non-stick pan, on low/med heat, slowly sweat mushrooms and scallions in half of the butter and 1/2 of the oil (oil will raise the smoke point of the butter to help prevent browning). Use a pinch of salt to help sweat the vegetables and shichimi to flavor them.
Once they have fully cooked and are somewhat dry, splash the sake and mix well with the saute. Alcohol in such small quantity is not an issue. Many foods have flavor compounds that are only soluble in alcohol so the sake helps release these flavors. Other mild liquors may be substituted. If alcohol is not an option, then any lightly sweet liquid will do. I have even used ginger ale. Set this aside along with any liquid in the pan.
In a bowl, beat the eggs with the water and add shichimi. Add remaining butter and oil to pan on low/med heat and pour eggs gently into the pan, then add salt to the eggs. Once the egg mixture has started to cook but is still mostly liquid, distribute sauted vegetables into the eggs. Doing this at this stage will keep the vegetables suspended in the eggs without being visible when the omelet is folded thus making a more pleasing presentation.
Continue cooking the eggs until almost cooked through. Using a lid at this stage can speed the cooking without having to turn up the heat.
Once the eggs are almost set, spread cream cheese on one side of the omelet and then the sliced tomato on the cream cheese. Just use enough slices needed to make a single layer on that half. Season tomato at this stage and then cook (again with lid) until cream cheese softens and the tomato warms a bit.
Flip the side without cream cheese and tomato onto the filled side and serve with lightly toasted crusty bread. The tomato usually produces a nicely flavored liquid so the bread comes in handy.