Preheat the oven to 375 degrees F (190 degrees C).
Heat olive oil in a saute pan over medium heat. Add onion; cook and stir until translucent, about 5 minutes. Add garlic and heat until fragrant, about 2 minutes. Remove from heat and let cool.
Combine chicken, onion mixture, cheese blend, cream cheese, salsa verde, olives, cilantro, and jalapenos in a large bowl for the filling. Mix well.
Melt butter in a saucepan over medium heat. Stir in flour, then chicken broth. Cook until sauce starts to thicken, 6 to 8 minutes. Stir in sour cream and salsa verde; cook until heated through. Spread a spoonful of sauce over the bottoms of two 9x11-inch casserole dishes.
Wrap a tortilla in a damp paper towel and microwave for about 30 seconds. Fill with the chicken mixture, roll up, and place seam-side down into a baking dish. Repeat with remaining tortillas until each dish holds 8 enchiladas.
Spoon remaining sauce over the enchiladas. Sprinkle cheese on top.
Bake in the preheated oven until heated through, about 20 minutes.
