Preheat oven to 350 degrees F. Spray an 8-inch or 9-inch square casserole dish with cooking spray and set aside.
Bring a pot of well-salted water to a boil. Add diced eggplant and cook just until tender, about 5-6 minutes. Drain well.
In a bowl, combine cracker crumbs, melted butter and grated cheddar cheese. Set aside.
Meanwhile, heat olive oil in a skillet over medium-high heat. Add the celery, onion and garlic to the skillet and sauté until tender, about 5-7 minutes. Combine celery mixture with the eggplant, milk, eggs, ½ teaspoon salt and ¼ teaspoon pepper. Gently fold in half of the cracker crumb mixture. Transfer to the prepared baking dish.
Top with the remaining half of the cracker crumb mixture.
Bake, uncovered, for about 25 minutes, or until the top is golden brown and the casserole is hot and bubbly. Allow the dish to rest for about 10 minutes before serving.