Melt 2 tablespoons butter in large skillet over medium-high heat. Add chicken and cook until no longer pink and starting to brown, about 8 to 10 minutes. Transfer chicken to large bowl, set aside.
Return the skillet to heat and add remaining 2 tablespoons butter. Add onion and jalapeños, cook until softened, about 5 minutes. Transfer vegetables to bowl with chicken; allow mixture to cool slightly. Add cream cheese, Monterey jack, and Mozzarella cheese, bacon, breadcrumbs, salt and pepper; mix until combined.
To assemble egg rolls, place one wrapper with point facing you. Place 2 tablespoons of filling in center of wrapper; spread mixture slightly to form a log. Fold one pointed side over filling; fold in ends and then roll to form tight log. Seal final edge with cornstarch mixture. Continue until all eggrolls are formed.
Fill large pot with oil at least 3 inches deep. Heat oil to 350 degrees F. Once hot, carefully drop egg rolls in a few at a time and fry, turning often, until brown and crispy, about 5 minutes. Do not overcrowd the pan or the oil will drop in temperature. Drain and transfer to a plate lined with paper towels. Serve egg rolls warm with ranch dressing for dipping.