For the Chicken: Combine first 9 ingredients in medium bowl; whisk to blend. Place chicken in 13 x 9 x 2-inch glass baking dish. Pour marinade over. Cover and refrigerate overnight, turning occasionally.
Prepare barbecue (medium-high heat). Remove chicken breasts from marinade. Grill chicken until just cooked through, turning occasionally, about 10 minutes. Place 1 cheese slice atop each chicken breast half; cover barbecue and cook until cheese melts, about 2 minutes. Transfer chicken breasts to plates. Serve with Ancho Chili Sauce.
Make the Sauce: Place chilies in medium metal bowl. Pour enough boiling water over chilies to cover. Let stand until chilies are soft, about 30 minutes. Drain, reserving 1/2 cup soaking liquid.
Puree chilies, 3 tablespoons soaking liquid and lime juice in blender until smooth. Transfer to small bowl. Whisk in mayonnaise, brown sugar, oregano, rosemary and cumin. Season to taste with salt and pepper. (Can be prepared 3 days ahead. Cover and refrigerate. Bring to room temperature before using, adding water by tablespoonfuls and stirring if very thick.).