Combine starter, rye flour, and one cup of the water in a large bowl; stir to blend; if you'd like an extra-sour flavor, cover mixture with plastic wrap and let stand in a warm place 6-24 hours (I let mine go six hours) until bubbly and sour, even boozy, smelling.
Combine yeast and remaining water in small bowl; let stand about 5 minutes until foamy; stir into sourdough mixture; add whole-wheat flour, 1 3/4 cups of the all-purpose flour, salt, sugar, and caraway seeds; stir (in KitchenAid stand mixer on low speed) until dough comes together.
Knead 9-10 minutes (in KitchenAid stand mixer on speed #2) until dough is smooth and springy; add more all-purpose flour, a tablespoon or two at a time during the kneading process as necessary to reduce excess stickiness (this particular dough will remain somewhat sticky); cover and let rise about 1 - 1 1/2 hours in a warm place until doubled.
Punch down dough and knead a few strokes to release air; shape into a round loaf and place on a baking stone or a greasing baking sheet; cover loaf lightly and allow to rise about 30 minutes until almost doubled.
Cut a couple small slashes about 3/4" deep on top of loaf; spray loaf with water and bake at 425° for 9 minutes, spraying loaf after 3, 6, and 9 minutes; reduce oven temperature to 400° and bake another 20-25 minutes until loaf is browned and sounds hollow when tapped on the bottom; transfer finished loaf to cooling rack.
NOTE: If you opt to let your starter and rye combination stand for several hours, you can get by without the additional yeast ~ Just plan on increased rising times.