In a large bowl combine the water, sugar, yeast, and 2 cups of the white whole wheat flour. Let the mixture rest for a few minutes, until bubbles form. Stir in the remaining 1 cup white whole wheat flour, the salt, and the oil., Stir in 2 cups of the all-purpose flour. Sprinkle an additional 1/2 cup flour over the sticky dough, mixing it in with a spatula or your hands., Sprinkle some of the remaining 1/2 cup all-purpose flour on a clean work surface. Place the dough onto the work surface and knead it for 5 minutes, adding a bit more flour if needed to prevent sticking., Alternatively, knead the dough in the bowl of your stand mixer; or in the bucket of a bread machine set on the dough cycle. Add the final 1/2 cup flour as needed to create a soft, smooth dough., Let the dough rest while you scrape out your mixing bowl and wipe it with oil., If you're kneading by hand, knead the dough a few minutes more; it should feel bouncy. Place it into the oiled bowl, and cover with plastic wrap., Place the dough somewhere warm (65°F to 75° is ideal) to rise until double in size, about 1 1/2 hours., Gently deflate the dough, and divide it in half. Shape each piece into an 8" to 9" log. Place each log into a lightly greased loaf pan, or onto a lightly greased or parchment-lined baking sheet., Tent the loaves with lightly greased plastic wrap, and let them rise for 30 minutes., While the loaves are rising, preheat your oven to 375°F., Uncover the bread, and bake it for about 30 minutes, until the crust is golden brown. If you have a digital thermometer, the internal temperature at the center will be about 190°F., Remove the bread from the oven, and transfer it to a rack to cool. Wrap completely cooled bread in plastic, and store at room temperature for several days; freeze for longer storage.