In a medium bowl, whisk together the eggs, Boursin cheese, milk, salt, and pepper until well combined and slightly frothy.
Heat a non-stick skillet over medium-low heat and add the butter, allowing it to melt and coat the pan.
Pour the egg mixture into the skillet and let it sit undisturbed for about 30 seconds until it begins to set around the edges.
Gently stir the eggs with a spatula, pushing them from the edges toward the center. Continue to cook, stirring occasionally, until the eggs are softly scrambled and slightly runny.
Remove the skillet from heat. The residual heat will continue to cook the eggs.
Serve immediately, garnished with chopped chives or parsley if desired.