1. Preheat oven to 450°F. Coat a baking sheet with cooking spray.
2. Toss zucchini, potato, 3 tablespoons shallot and egg in a large bowl. Add breadcrumbs, feta, 1 tablespoon dill, 1/4 teaspoon salt and 1/4 teaspoon pepper; toss to combine. Form the mixture into 12 patties.
3. Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add 6 patties, cover and cook until crispy and browned on one side, 2 to 5 minutes. Carefully transfer the latkes to the prepared pan, browned-side down. Repeat with the remaining 1 tablespoon oil and patties.
4. Transfer the latkes to the oven and bake until firm and heated through, 10 to 12 minutes.
5. Meanwhile, prepare tzatziki: Combine yogurt, cucumber, vinegar, the remaining minced shallot, 1 tablespoon dill and 1/4 teaspoon each salt and pepper in a small bowl. Serve the latkes with the tzatziki on the side.