Combine 2-1/4 cups flour, sugar, salt, dill and undissolved yeast in a large mixing bowl. Stir in water and butter; mix in enough remaining flour to make a soft dough. Turn out onto lightly floured surface and knead 4 minutes. Cover and let rest 10 minutes.
Roll dough into a 12 x 10-inch rectangle. Mix tuna, mayonnaise and onion; spread evenly over dough leaving 1/2-inch around edges. Sprinkle with cheese. Roll up from the long side, pinching seam and ends to seal. Place on greased baking sheet. Cut 3 or 4 slits in the top. Cover; let rise in a warm draft-free place for 15 minutes.
Bake in preheated 400°F oven for 30 minutes or until done. Let stand 5 minutes before slicing; serve warm.