This gorgeous side goes well with simply grilled meat or poultry. Yellow squash has a high water content, which means that cooking the squash noodles would release a lot of liquid. We opt instead to air-dry them and keep them raw for more freshness and plenty of crunch. Look for lemons with thinner skins (smoother, shinier skins are often thinner); you'll get a nice citrus hit with each bite, but charring the slices softens the flavor so it's not overpowering. Serve right after tossing everything together, as the squash will start to wilt.
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.