Activate the Yeast: In the bowl of a stand mixer fitted with the dough hook, whisk together the warm water, honey, and active dry yeast until combined. Let the mixture stand for 5 minutes, until it becomes very foamy. Form the Dough: Add the bread flour, rye flour, vegetable oil, caraway seeds, and salt to the yeast mixture. Stir with a wooden spoon or rubber spatula until all of the dry flour from the bottom and sides of the bowl has been incorporated and a shaggy dough forms. Mix the Dough: Using the stand mixer, mix the dough on medium-low speed for 6 to 8 minutes. Continue until most of the dough gathers around the dough hook and the dough becomes smooth while remaining slightly sticky. Prepare for the First Rise: Transfer the dough to a clean, lightly oiled bowl. Turn the dough over once so its surface is lightly coated with oil.Scrape down the sides of the bowl and shape the dough into a ball. Let the Dough Rise: Cover the bowl with plastic wrap and place it in a warm, draft-free spot at about 75°F. Let the dough rise for 30 to 45 minutes, or until it has doubled in size. Prepare the Loaf Pan: While the dough rises, grease a 9×5-inch loaf pan with oil or baking spray. Set the prepared pan aside. Shape the Loaf: Turn the risen dough onto a very lightly floured surface. Using moist hands, stretch and shape the dough into a 9-inch square. Roll the Dough: Use a bench scraper to help loosen and coax the dough from the work surface as you roll it into a log. Pinch the seams along the sides firmly closed, then gently tuck the seams underneath the loaf. Arrange in the Pan: Place the shaped dough into the prepared loaf pan with the seam side facing down.
Cover for the Second Rise: Lightly oil a piece of plastic wrap or coat it with nonstick spray. Cover the loaf pan and return it to a warm, draft-free spot at about 75°F. Complete the Second Rise: Let the dough rise for 35 to 45 minutes, or until it has risen about ½ to 1 inch above the rim of the loaf pan.
Preheat the Oven: After the dough has been rising for 15 minutes, preheat the oven to 350°F.
Bake the Bread: Bake the loaf for 35 to 40 minutes, until the top is golden brown and the bread sounds hollow when tapped.
Cool the Loaf: Let the bread cool in the pan for a couple of minutes, then turn it out onto a wire rack.Allow it to cool completely before slicing so the crumb can properly set.