In a 12-inch nonstick skillet over medium, melt 2 tablespoons butter. Stir in the wine and mustard. Simmer, stirring, until the liquid evaporates and the mixture sizzles, 2 to 3 minutes. Add the bread, toss to coat, then cook, stirring, until golden brown and crisp, 3 to 4 minutes; transfer to a plate. Wipe out the skillet and let cool. In a medium bowl, whisk together the eggs, 1 tablespoon chives and ¼ teaspoon each salt and pepper.

Return the skillet to medium-low, add and melt the remaining 1 tablespoon butter. Add the egg mixture. Working slowly around the pan, use a silicone spatula to gently draw the edges inward, tilting the pan to fill the space with egg, until the eggs no longer are runny and have formed a ruffled-edged omelet with a shiny, wet surface, about 3 minutes.

Off heat, scatter the cheese over the eggs. Cover and let stand until melted and the surface of the omelet is set, about 3 minutes. Run the spatula around the edge and underneath to loosen. Scatter the toasted bread over half the omelet, then use the spatula to fold the other half over it. Slide onto a serving plate and sprinkle with the remaining 1 tablespoon chives.