You can mix whole-wheat pasta dough without a food processor. You just need to get your hands dirty.
Whisk 1½ cups (8¼ ounces) whole-wheat flour and ½ cup (2½ ounces) all-purpose flour in large bowl. Stir in 3 large eggs, lightly beaten, and 1½ tablespoons water into flour mixture. Using your hands, knead dough in bowl until mixture forms cohesive dough that feels soft and is barely tacky to touch, about 3 minutes. (If dough sticks to fingers, add up to ¼ cup all-purpose flour, 1 tablespoon at a time, until barely tacky. If dough doesn’t become cohesive, add 1 tablespoon water, 1 teaspoon at a time, until it just comes together; knead 1 minute longer.)
Transfer dough to clean surface and knead by hand to form smooth, uniform ball, 1 to 2 minutes. Shape dough into ball. Wrap with plastic wrap and set aside at room temperature to rest for at least 1 hour or up to 4 hours.