Stir white sugar, 1/4 cup butter, and salt into hot milk in a small bowl until butter is melted and mixture is lukewarm. In the work bowl of a stand mixer fitted with a paddle, sprinkle yeast over warm water; stir to dissolve.
Stir milk mixture into yeast mixture. Add 2 eggs and 2 1/2 cups of the flour; beat on high until combined. With mixer on low, add remaining 2 cups flour, 1/2 cup at a time.
Switch to the dough hook; mix until dough leaves sides of bowl, about 5 minutes. Set dough in a large, lightly greased bowl, turning dough to coat top. Cover with a towel and let rise in a warm spot until doubled, about 1 hour.
Prepare walnut filling while dough rises. Lightly beat 3 eggs in a large bowl. Stir in brown sugar, 1/3 cup melted butter, cinnamon, and vanilla extract. Fold in walnuts.
Punch dough down. Turn dough out onto a large, lightly floured surface, cover with the bowl, and let rest 10 minutes. Line three 9×5-inch loaf pans with parchment paper, leaving a 2-inch overhang on long sides.
Divide dough into thirds; roll each third out to a 12-inch square on a lightly floured surface. Spoon 1/3 of the walnut filling onto each dough square and spread to within 1/2 inch of the edges. Roll up each square tightly as for a jelly roll. Pinch ends and seams closed to seal. With your palms, roll logs back and forth until uniformly round.