Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Beat the tCream Cheese (2 ounce) in a medium bowl with an electric mixer for about 30 seconds. Gradually beat in Milk (1/4 cup) until smooth.
Beat in the Egg (4), Salt (1/8 teaspoon), and Freshly Ground Black Pepper (1/8 teaspoon).
Melt the Butter (1 tablespoon) in a medium skillet over medium heat. Add the egg mixture and cook, stirring, until the eggs are scrambled and nearly cooked through, for about 3 to 4 minutes.
Unroll the Crescent Dough Sheet (1 package) onto the prepared baking sheet. With the short side of the dough closest to you, cut 12 horizontal slits on each of the outer two-thirds of the dough, leaving the center one-third uncut.
Spread the Dijon Mustard (1 tablespoon) evenly along the center of the dough with a pastry brush.
Next, add an even layer of Deli Ham (2 ounce), then the scrambled eggs and Shredded Cheddar Cheese (1/2 cup).
Crisscross the dough flaps over the filling to form a braid pattern.
Bake until golden brown, 20 to 25 minutes.
Garnish with the Fresh Chives (to taste). Serve immediately.