In a pan, combine chicken with chicken broth, onion, 2 cloves of garlic, and 2 teaspoons salt. ( I usually like to add 1/2 Tablespoons of chicken bouillon to add extra flavor to the chicken). Boil for 20 minutes. Reserve broth, set chicken aside to cool, and discard onion and garlic. Bone and shred chicken when cool. Set shredded chicken aside.
Discard husks and stems from tomatillos. Place tomatillos and serrano chiles in a pot with water, enough just to cover them. Bring to boil, and continue boiling until tomatillos turn a different shade of green (from bright green to a dull, army green). Strain tomatillos and chiles, and place in a blender with the other half of onion, 2 cloves garlic, cilantro, and 1 teaspoon of salt. Pour in reserved chicken broth, so that liquid just covers the veggies in the blender by about an inch. Blend all ingredients until they are completely pureed. Pour salsa in a medium saucepan, and bring to a low boil. Let it cool down a bit.
In a bowl, add half of the salsa verde to the shredded chicken. Taste and adjust seasonings (salt and pepper) if neccessary. Set aside.
Pour oil in a frying pan, and allow to get very hot. Slightly fry tortillas one by one in hot oil, setting each on a paper towel afterwards to soak some of the oil. (To save time and calories, you can brush the tortillas with oil and place them in a plastic tortilla warmer or wrap them n a towel, and microwave them at high for 30-45 seconds, or until they are hot and pliable.)
Spread 1/2 cup salsa verde in bottom of a a 13x9x2-inch baking dish.
Assemble the enchiladas by spooning about 1/3 cup of the chicken filling mixture in each tortilla, add grated cheese. Roll up tortilla and place in the baking dish. Continue with the rest of the tortillas. Arrange them seam sides down, crosswise in dish. Pour the remaining salsa over the top of the tortillas and sprinkle with remaining cheese and chopped cilantro. Cover and bake at 400 degrees for 10 minutes or until thoroughly heated and cheese melted. Serve with sour cream.