Step 1Combine chicken and broth in bowl. Toss to moisten meat. Add 1/3 cup shredded cheese. Toss again. Season with salt and pepper to taste.
Step 2Heat oil in medium nonstick skillet over high heat. When very hot, carefully working with 1 tortilla at a time, use tongs to quickly dip both sides of tortilla in oil (don't let them cook). Blot on paper towels. Stack hot tortillas and wrap in foil to keep warm.
Step 3Put about 1/4 cup chicken along center of each heated tortilla; roll up tightly. Place seam side down in single layer in greased 13x9-inch baking dish. Pour salsa verde evenly over tortillas. Sprinkle remaining cheese over top. (Can be made ahead and refrigerated overnight, covered airtight.)
Step 4Bake at 375 degrees, covered, until very hot and cheese is melted, about 30 minutes. Uncover and bake 5 minutes more. Serve hot. Pass salsa, hot sauce and sour cream.