Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add chayote to pan; sauté 10 minutes or until crisp-tender. Reduce heat to low; add 1/4 cup water. Cover and cook 10 minutes or until tender. Place chayote in a large bowl; mash with a potato masher. Drain chayote through a fine sieve for 30 minutes, reserving chayote pulp.
Preheat oven to 350°.
Heat oil in a large nonstick skillet coated with cooking spray over medium-high heat. Add onion, celery, pepper, and thyme to pan; sauté 10 minutes or until vegetables are tender. Add chayote pulp and salt, stirring to combine. Spoon mixture into an 8-inch baking dish coated with cooking spray.
Place bread in a food processor; pulse 10 times or until fine crumbs measure 1/2 cup. Combine crumbs and cheese in a small bowl; stir in butter. Sprinkle crumb mixture evenly over chayote mixture. Bake at 350° for 25 minutes or until golden.