Stir together flour, baking soda, baking powder, salt, and sugar in a large bowl. Make a well in the center.
Pour eggs and 1 cup milk into well; stir to blend, adding more milk to make a moist, battery dough, unlike pancake batter. Use a sturdy wooden spoon to beat dough for at least 200 strokes, rolling it over and over in the bowl until smooth and an occasional bubble appears on the surface. Add dry white bread cubes; stir into dough until they disappear.
Bring a large pot of water to a boil. Place dough onto a cheesecloth or white cotton cloth; form into a loaf shape. Wrap the cloth around loaf and tie ends.
Place loaf into boiling water; cook 45 minutes, turning loaf over halfway through. Remove from water, unwrap, and cover with a tea towel. Let stand for 10 minutes.
Fry bacon in a small skillet over medium-high heat until evenly browned. Set aside.
Place drained sauerkraut into a saucepan; add water to cover the surface. Bring to a simmer over medium heat. Add bacon; season with salt, pepper, and caraway seeds. Stir together water and cornstarch; mix into sauerkraut and simmer for a few minutes before removing from heat.
Slice dumpling loaf. Drizzle with bacon drippings from the pan. Serve with sauerkraut.