Trim squash ends, and halve squash crosswise. Using a spoon, carefully scoop out inner seeds from each squash, removing as little flesh as possible. Cut squash into large noodles using a spiralizer. Place squash noodles between 2 layers of paper towels. Let stand at room temperature until dry, about 3 hours.
Transfer noodles to a medium bowl. Grate rind from 1 lemon to equal 1/2 teaspoon. Cut same lemon in half; squeeze juice to equal 1 tablespoon. Set aside rind and juice. Trim tops and bottoms of remaining 2 lemons, and slice into half-moons.
Heat 1 1/2 teaspoons oil in a large nonstick skillet over medium-high. Add lemon slices; sprinkle evenly with 1/8 teaspoon salt. Sprinkle sugar over lemons. Cook, stirring occasionally, until lemons are browned on the edges and caramelized, 4 to 5 minutes.
Add caramelized lemon slices to squash noodles. Add celery leaves, cheese, lemon rind, lemon juice, pepper, and remaining 1/8 teaspoon salt; toss gently to combine. Drizzle with remaining 1 tablespoon oil. Serve immediately.