Preheat oven to 375°F. Line a baking sheet with parchment paper.

Beat the cream cheese in a medium bowl with an electric mixer for about 30 seconds. Gradually beat in milk until smooth. Beat in the eggs, salt & pepper.

Melt the butter 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 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 evenly along the center of the dough with a pastry brush. Next, add an even layer of ham, then the scrambled eggs and shredded cheddar cheese.

Crisscross the dough flaps over the filling to form a braid pattern. Bake until golden brown, 20 to 25 minutes. Garnish with the snipped fresh chives. Serve immediately.