Melt butter in a saucepan. Stir in flour and cook for 2 minutes. Add milk and bring to a slow boil, stirring frequently. Let slowly boil until thickened to about a canned cream-of soup consistency.
Remove from heat and stir in cheese until melted. Stir in remaining ingredients one at a time in order listed until well combined.
To Assemble:
Divide the dough into 12 equally sized balls. Roll each ball on a floured surface into a circle. Place about a 1/4 cup chicken and broccoli filling in the center.
Bring the sides up and pinch together tightly to form one seal along the top. Tuck that seam under itself and turn over. Shape into a rounded roll shape and place on a baking sheet. Repeat until done. If desired, lightly brush the rolls with egg white.
Bake rolls at 375 degrees for about 15-20 minutes, until lightly golden brown and hollow sounding when tapped.