Fill a medium pot with an inch of water, then add eggplant and 1/2 teaspoon salt. Cover and bring to a boil over medium heat. Cook until eggplant is tender, about 15 minutes. Drain well.
Transfer eggplant to a large bowl; stir in 3/4 cup bread crumbs, bell pepper, onion, Cheddar, one egg, remaining 1 1/4 teaspoons salt, paprika, and pepper until well combined. Cover and refrigerate until chilled, at least 2 hours or overnight.
When ready to cook, set out three shallow bowls. Place flour in one bowl, beat remaining egg in another bowl, and place remaining 3/4 cup bread crumbs in the third bowl.
Drop heaping teaspoonfuls of eggplant mixture into flour; roll until completely coated. Dip into beaten egg, then lift up so excess egg drips back into the bowl, and roll in bread crumbs to coat.
Heat oil in a large saucepan to 350 degrees F (175 degrees C).
Fry eggplant balls in hot oil until golden brown and heated through, 2 to 3 minutes per side. Drain on paper towels.