2. Meanwhile, to make the dough, process 1/2 cup of the walnuts and the flour in a food processor until the walnuts are ground into a powder. Set aside.
3. Add the cool water, walnut oil, and salt to the mixer bowl and stir well. Attach the bowl to the mixer with the paddle blade. Mix on low speed, adding enough of the walnut flour to make a soft, pliable dough that barely clears the sides of the bowl. Switch to the dough hook and mix on medium-low speed until the dough is smooth and elastic, about 10 minutes. During the last minute, add the remaining 1/2 cup of walnuts to mix them in.
4. Gather up the dough into a ball. Lightly oil a medium bowl. Add the ball of dough and turn to coat the dough with oil. Cover tightly with plastic wrap and let stand until the dough doubles in volume, about 1 1/2 hours. (If you have the time, refrigerate the dough for 8 to 18 hours. Remove from the refrigerator and let come to room temperature before proceeding.)
5. Position a rack in the center of the oven and preheat to 350°F. Turn out the dough onto an unfloured work surface, and knead lightly to expel the air. Cut the dough into 12 equal pieces. Shape each piece into tight balls. Arrange the balls a few inches apart, smooth sides up, on an ungreased baking sheet. Cover loosely with plastic wrap and let stand until almost doubled, about 30 minutes.
6. Discard the plastic wrap. Bake the rolls until crusty, about 20 minutes. To double-check for doneness, insert an instant-read thermometer into the center of a roll--it should read about 200°F. Cool slightly or to room temperature.