Heat oven to 400°. Place cod on a rimmed baking sheet. Rub with 1 Tbsp. oil. Roast until just opaque in center, 10–15 minutes. Let fish cool, then flake with a fork.
Mix cod, whitefish, eggs, shallot, matzo, chives, 2 Tbsp. dill, and 2 Tbsp. water in a medium bowl. Season with pepper and mix vigorously to combine (mixture will be stiff).
Reduce oven temperature to 200°; set a wire rack inside a large rimmed baking sheet. Heat remaining 4 Tbsp. oil in a large skillet over medium-high heat. Working in batches, drop scant ¼-cupfuls of fritter mixture into skillet; press gently with a spatula to flatten. Cook, turning once and adding more oil if needed, until golden brown and crispy around edges, about 4 minutes per batch. Transfer to prepared rack. Keep warm in oven while cooking remaining fritters.
Mix beets into vinaigrette. Serve fritters with vinaigrette; top with dill.
DO AHEAD: Fritter mixture can be made 1 day ahead; cover and chill.