Preheat oven to 400° with a rack in lowest position. In a large frying pan over medium heat, sauté onions with 2 tbsp. oil and the salt, covered, until softened but not browned, about 10 minutes.
Whisk eggs and milk in a bowl. Stir in cheeses, caraway, lemon juice, dill, semolina, pepper, and 2 tbsp. oil. Stir in onions.
Oil a 12-in. tart pan with removable rim. Transfer 1 filo sheet from package to a work surface; cover remaining filo with a damp kitchen towel. Brush filo sheet with oil and ease into tart pan with at least 2 in. hanging over sides. Repeat with 3 more filo sheets, rotating each 45° to cover pan bottom and sides with plenty of overhang (they'll overlap). Spoon filling into tart.
Oil sheet 5 of filo and lay on top of filling, rumpling filo artfully over entire top and especially in the center. Top with sheet 6, rumpling it the same way. Fold overhanging filo back over tart (it will break a little) and dab any dry spots with oil.
Bake tart until golden brown, about 50 minutes. Let cool 20 minutes, then free tart edges from pan with a small spatula. Set tart on a platter. Cut in wedges and serve warm or at room temperature.
Make ahead: Up to 8 hours; chill uncovered, then reheat in a 350° oven until warm and crisp, about 20 minutes.
Note: Nutritional analysis is per serving.