Heat the milk in a small saucepan until it is just warm. Pour it into a very large glass or ceramic bowl and stir in the yeast, 1/2 cup of the flour, and 6 tablespoons of the sugar. Place the bowl in a warm, draft-free place, lightly covered with a cloth, and allow to stand until the yeast starts to bubble up, about 30 minutes.
With an electric mixer in a medium bowl, beat the egg yolks with 6 tablespoons of the remaining sugar until pale and creamy. Pour this mixture into the yeast mixture. Stir in the drained orange or bergamot spoon sweet, vanilla, kumquat or orange liqueur, and melted butter. Slowly add flour, 1 cup at a time, mixing it in with a wooden spoon, until a mass begins to form. Turn this out onto a lightly floured work surface and knead, adding more flour if necessary to form a very smooth, large ball of dough. Cover, place in a warm, draft-free place, and allow to stand until the dough has doubled in bulk, about 2 hours.
Punch the dough down and knead again for about 7 to 8 minutes. Shape into 1 round loaf and place on a buttered baking sheet. Make 2 sharp slashes like an X on the surface of the dough, cover with a cloth, and let rise again until doubled for about 1 hour.
Preheat oven to 350 degrees. With an electric mixer, beat the egg whites with remaining 1 tablespoon sugar in a medium bowl until they are frothy without forming peaks. Brush the surface of each loaf with the mixture, sprinkle with caraway seeds, and bake until the fogatsa has swelled and browned, about 1 hour. Remove from the oven, let cool on a wire rack, and serve.