Preheat oven to 375°F. Chop chicken and broccoli and place in a 2qt bowl. Chop bell pepper and add to the bowl. Press garlic over vegetable mixture. Add shredded cheese and mix gently. Add mayonnaise, dill weed and salt. Mix well.
Unroll 1 pkg. of crescent dough, do not separate. Arrange longest sides of dough across width of 12" x 15" pan. Repeat with remaining package of dough. Press seams of dough to seal perforations. On longest sides of the pan, cut dough into strips 1 1/2 inches apart, 3 inches deep. Spread filling evenly over middle of dough. To braid, lift strips of dough across mixture to meet in center, twisting each strip one turn. Continue alternating strips to form a braid. Tuck ends up to seal at end of braid. Brush egg white over dough.