Bring a small pot of salted water to a boil. Add the whole chicken breast and cook until no longer pink inside, about 10–15 minutes. Remove, let cool slightly, and shred with two forks into about 1 cup shredded chicken.
Preheat the broiler to HIGH and move the oven rack to the second position from the top. Lightly spray a baking sheet with avocado oil and set aside.
Heat a small nonstick skillet over high heat and lightly spray with avocado oil. Once the pan is hot, reduce heat to low. Pour about 1/2 cup liquid egg whites into the pan and cook until set, about 4–5 minutes, then carefully lift the egg "tortilla" onto the prepared baking sheet. Repeat to make four egg tortillas from the remaining egg whites.
Place the shredded chicken in a mixing bowl and stir in 1/4 cup BBQ sauce until evenly coated.
Assemble two quesadillas on the baking sheet: divide the BBQ chicken evenly between two egg tortillas, sprinkle each with 1/3 cup grated cheddar and 1 tablespoon chopped cilantro, then top each with a second egg tortilla. Lightly spray the tops with avocado oil.
Broil the assembled quesadillas on the prepared rack until the tops are lightly browned and the cheese is melted, about 15 minutes. Remove from the oven and let cool briefly before slicing and serving. Drizzle additional BBQ sauce if desired.
