Place 4 cups water, broccoli, and carrots in a small saucepan; bring to a boil. Cook 4 minutes; drain. Pat broccoli mixture dry with paper towels; finely chop. Place broccoli mixture and flour in a large bowl; stir to coat. Add cheese, salt, pepper, onions, and egg to broccoli mixture; stir to combine.
Heat a large nonstick skillet over medium-high heat. Add oil to pan; swirl to coat. Spoon 1/4 cup broccoli mixture into a dry measuring cup. Pour mixture into pan; flatten slightly. Repeat procedure 7 times to form 8 fritters. Cook 4 minutes on each side or until golden brown. Combine yogurt and dill in a small bowl. Serve yogurt mixture with fritters.