Melt butter in heavy large skillet over medium heat. Add onion and saute until tender, about 5 minutes. Let cool. Combine milk and sugar in large bowl. Sprinkle yeast over; stir to blend. Let stand until foamy, about 10 minutes. Transfer to processor. Mix in onion mixture, salt and 1 cup flour. Blend in enough remaining flour, 1 cup at a time, to form smooth ball. Transfer dough to floured work surface and knead until smooth and elastic, about 3 minutes. Shape dough into ball. Butter large bowl. Place dough in bowl; turn to coat. Cover bowl with clean kitchen towel. Let dough rise in warm draft-free area until doubled in volume, about 45 minutes. Butter heavy large baking sheet. Punch down dough. Turn out onto floured work surface. Sprinkle walnuts over; knead to incorporate. Divide dough in half. Shape each half into 6-inch round loaf. Place dough rounds on prepared baking sheet, spacing 4 inches apart. Flatten slightly. Cover with dry towel. Let rise in warm draft-free area until almost doubled in volume, about 45 minutes. Preheat oven to 400°F. Bake until loaves are golden brown and sound hollow when tapped on bottom, about 25 minutes. Transfer to racks; cool. Serve at room temperature.