Cut Queso de Freir lengthwise into four ½-inch-thick slices. Preheat waffle iron; spray with non-stick cooking spray. Place a cheese slice in center of waffle iron. Close waffle iron; cook for 2-3 minutes until cheese is golden brown. Remove from waffle iron. Repeat with remaining cheese slices.
Cook bacon slices until crisp. Cook eggs over-easy.
To serve, place 1 egg on cheese slice; top with bacon and avocado slices. Top with cheese slice. Repeat with remaining ingredients to make 2 sandwiches.