Whisk the eggs with the cream and black pepper until uniformly yellow, but do not beat air into them. Omit salt: the smoked chicken is already salty enough, and adding salt to raw eggs too early draws moisture from the yolks, which can cause the omelet to weep or curdle later.
Stir the cream cheese with finely chopped tarragon, some lemon zest, and a few drops of lemon juice until smooth and spreadable. Tarragon has a distinct anise-like note that provides just the right counterpoint to the richness of the cream cheese when paired with egg and poultry.
Cut the smoked chicken breast into thin strips.
Melt the butter in a frying pan of about twenty-four centimeters over medium-low heat. Pour in the egg mixture as soon as the butter foams but before it colors, tilting the pan to distribute it evenly. Keep the heat low: an omelet cooked too quickly becomes brown and rubbery, whereas low heat keeps it silky smooth.
Gently lift the edges with a spatula and let the still-liquid egg run underneath. Remove the pan from the heat as soon as the top is just barely glossy; the residual heat will finish cooking it in half a minute.
Spread the tarragon cream cheese over half of the omelet, top with the chicken strips, and fold it over.
Slide it onto a plate with the watercress on the side, and finish with finely snipped chives and black pepper.