Place chicken breasts in medium size saucepan; add water and sliced garlic. Cover and bring to a simmer. Cook just until tender, about 15 minutes. Cool. Remove chicken and reserve the broth. Cut chicken into thin julienne strips. Set aside in a bowl.
Saute onion in butter in medium size skillet just until soft, about 5 minutes. Add chopped garlic; saute 1 minute. Add chilies, chili powder, cumin, salt, oregano, and pepper. Cook 1 minute. Stir in flour until well combined. Cook 1 minute, stirring.
Stir in reserved chicken-cooking broth, the 1 cup chicken broth and heavy cream. Cook over medium heat, stirring frequently until mixture thickens, about 10 minutes. Remove from heat. Stir in 1 cup of shredded cheese until melted.
Combine 1 cup cheese sauce with the reserved chicken. Preheat oven to 400 degrees. Heat vegetable oil in small skillet until hot. Dip tortillas, one at a time, in hot oil just until limp, about 5 to 10 seconds on each side. Do not let them become crisp. Place on work surface.
Divide chicken filling equally along the center of each tortilla. Top each with sliced green onions. Roll up each tortilla and place, seam side down, in 2 rows in 13 x 9 x 2 inch baking dish. Pour remaining cheese sauce evenly over tortillas. Sprinkle with remaining 1 cup cheese. Bake in preheated hot oven (400 degrees) for 20 minutes or until bubbly. Garnish with extra sliced green onion if you wish.
