Preheat the oven to 450 degrees F. Melt the butter with 2 tablespoons vegetable oil in a skillet; transfer to a bowl. Heat the remaining 2 tablespoons oil in the skillet. Add the onion, garlic and jalapeno and cook until soft, about 3 minutes. Add the chili powder, cumin, cinnamon and 1 teaspoon salt; toast 30 seconds. Add the tomato and cilantro and cook until slightly dry, about 2 minutes. Stir in the chicken and sour cream and warm through. Remove from the heat.
Brush a rimmed baking sheet with some of the butter-oil mixture. Spread 2 tablespoons refried beans down the center of each tortilla, leaving a 2-inch border on both ends. Top with 1 cup chicken mixture and 1/4 cup cheese. Fold in the ends and roll up.
Put the chimichangas seam-side down on the baking sheet; brush with the butter-oil mixture. Bake 8 to 10 minutes per side, brushing again after you flip. Top with the sauce, more cheese, lettuce and tomato. Serve with rice and the remaining beans.
Make Chi-Chi's Mexi-sauce:
Saute 1/2 cup chopped onion and 2 chopped garlic cloves in a skillet with vegetable oil. Add a pinch each of chili powder, cumin, sugar and salt; cook 30 seconds. Stir in two 4-ounce cans chopped green chiles (drained and rinsed); cook 2 minutes. Add 1 cup chicken broth and simmer until thickened, then puree. Stir in 1/4 cup chopped cilantro.