Season chicken with salt and pepper on both sides.
Heat oil in a medium skillet over medium heat. Add the chicken and cook turning once or twice and adjusting the heat to prevent burning, until brown and an instant-read thermometer inserted into the thickest part registers 165 degrees, 12 to 16 minutes. Transfer to a plate and tent with foil to keep warm.
Add shallot to the pan and cook, stirring, until fragrant, about 30 seconds. Add mushrooms, cook, stirring occasionally, until tender, about 2 minutes. Pour in vermouth (or wine); simmer until almost evaporated, scraping up any browned bits, about 1 minutes. Pour in broth and cook until reduced by half, 1 to 2 minutes. Stir in cream and chives (or scallions); return to a simmer. Return chicken to the pan, turn to coat with sauce and cook until heated through, about 1 minute.Per serving: 275 cal.; 25 g pro.; 5 g carb.; 15 g fat (5 sat., 7 monounsat., 3 polyunsat.); 84 mg chol.; 373 mg sod.; 1 g fiber; 49 percent calories from fat.
In a small bowl combine eggs, water, salt and pepper. Beat until combined, but not frothy, with a fork. Heat an 8-inch nonstick skillet with flared sides over medium-high heat until skillet is hot.
Add butter to skillet. When butter has melted add egg mixture to skillet, lower heat to medium. Immediately begin stirring the eggs gently but continuously with a wooden or plastic spatula until mixture resembles small pieces of cook egg surrounded by liquid egg. Stop stirring. Cook for 30 to 60 seconds more or until egg is set but shiny.
Spoon shredded cheese across the center. With a spatula lift and fold an edge of the omelet about a third of the way toward the center. Remove from the heat. Fold the opposite edge of omelet toward the center and transfer to a warm plate.Per serving: 246 cal.; 13 g pro.; 1 g carb.; 21 g fat (11 sat., 7 monounsat., 3 polyunsat.); 403 mg chol.; 434 mg sod.; 0 g fiber; 77 percent calories from fat.