Remove the skin and bones from the chicken and shred into a large mixing bowl. Toss together with 1 cup barbecue sauce, cheeses, and 1/4 cup scallions.
To assemble the egg rolls, working one at a time, place the egg roll wrapper at an angle on a clean work surface. Spoon about 2 tablespoons of the chicken mixture in the center. Top with a few slices of red onion and cilantro leaves.
Bring the corners into the center, over the filling, and roll into a tight log. Seal the edge with a bit of beaten egg and place on a platter or baking sheet. Continue rolling the rest of the egg rolls in the same manner.
Fill a pot or skillet with high sides, halfway up with oil and bring to 365 degrees F. Once hot carefully add a few egg rolls at a time. Fry until golden brown, about 2-3 minutes per side. Drain on paper towels and serve with remaining BBQ sauce and garnish with cilantro and sliced scallions.