With a sharp knife, 3/4 to 1 inch in from crust rim, cut down to but not through bottom crust. Then cut a circle inside rim of loaf.
With your fingers, reach down into the cut and pull out center of loaf. Then make the base of the hollow level by pulling out excess bread.
Brush the interior of the loaf with enough butter to lightly coat. Wrap the crust with foil.
Bake in a 350° oven until interior is lightly toasted, 7 to 9 minutes.
Meanwhile, in a food processor or blender, combine yogurt cheese, eggs, 1/2 cup milk, flour, ground chili, and 1/4 teaspoon salt. Whirl until smooth. Stir green chilies into cheese mixture.
Set hot crust on a baking sheet and pour filling into it. Add remaining 1/4 cup milk, stir with a fork, then push enough 1/2-inch chunks of bread (torn from the center) into the filling to bring it up to the crust rim.
Return loaf to oven. Bake until center is firmly set when loaf is gently shaken, 40 to 45 minutes. Remove loaf from oven and immediately sprinkle with cheddar cheese.
Let cool at least 15 minutes. Remove foil. Peel and thinly slice or dice avocado, mix with lime juice, and arrange on chili loaf. Cut loaf and serve with tomatillo sauce and salt to taste.