Heat oil in a large heavy-bottomed skillet over medium-high heat until shimmering, about 350° to 375° on a deep-fry thermometer. Fry tortilla strips in 3 batches until golden-brown, 1 to 2 minutes each, gently moving around in oil with a slotted spoon. Remove with spoon to a paper-towel-lined plate. Season strips with 1/4 tsp of the salt.
Carefully discard all but 1 tbsp of the oil. Add eggs and cook over medium heat until scrambled, about 2 minutes. Season with remaining 1/8 tsp salt. Meanwhile, heat chicken and salsa verde in a medium pan.
Arrange tortilla strips on a large platter. Spoon chicken mixture on top, then scatter eggs on the chicken. Garnish with shredded queso fresco, sliced scallions and chopped cilantro, if desired.