Preheat oven to 350 degrees F. In a medium bowl, whisk the chickpea flour with the water, 1 egg, and salt until no lumps remain.
Heat a large nonstick skillet over medium-high heat. Lightly brush the skillet with oil. Add ⅓ cup of the batter, swirling the skillet a few times to fully coat the edges. Cook until golden, about 1 to 2 minutes. Using a spatula, flip the crepe and cook until golden on the other side, about 1 to 2 minutes more. Transfer to a plate and repeat with the remaining 3 crepes.
Place the crepes on a rimmed baking sheet. Place cheese, if using, and a slice of turkey on each crepe. Crack 1 egg into the center of each crepe; folding the edges toward the center to enclose the egg. Season with salt and pepper. Bake until egg white is set and yolk is still runny, about 12 minutes. Top with chives.