For filling: Place chicken and broth in large saucepan. Add carrots, onion, garlic, parsley and bay leaves. Bring to boil. Cover and simmer 5 minutes. Turn off heat and let stand until chicken is cooked through, about 1 hour. Remove chicken from broth and cool slightly. Strain cooking liquid. Return to saucepan and boil until reduced to 3 cups. skin and bone chicken. Shred meat and place in medium bowl. Chill.
For sauce: Melt butter in heavy medium saucepan over how heat. Add flour and cook until light brown, stirring frequently, about 5 minutes. Whisk in reserved chicken cooking liquid. Increase heat to medium and cook until sauce thickens, stirring occasionally. Cool.
Heat 1 Tbl oil in heavy large skillet over medium heat. Add onions and cook until soft, stirring occasionally, about 5 minutes. Add spinach and stir until wilted, about 2 minutes. Transfer to processor. Add sauce, 1 1/2 cups sour cream, green onions and 1 can chilies and puree until smooth. Season with salt and pepper. Add second can of chilies if desired.
Butter 2 9x13-inch baking dishes. Spread 1/2 cup sauce over bottom of each. Add 1 cup sauce to chicken. Mix in 5 cups grated cheese. Pour 1 inch oil into heavy medium skillet. Heat over medium-low heat. Add 1 tortilla and cook until softened, about 5 seconds. Drain on paper towels. Repeat with remaining tortillas. Divide chicken filling among tortillas and roll up. Arrange seam side down in prepared dishes. Spoon remaining sauce over. Cover with foil. (Can be prepared 1 day ahead. Refrigerate enchiladas and remaining cheese separately. Bring enchiladas to room temperature before baking).
Preheat oven to 400°F.
Bake covered enchiladas until just heated through, about 10 minutes. Uncover, sprinkle with remaining cheese and bake until cheese melts, about 5 minutes. Serve with sour cream.