Put flours, yeast, sugar and salt into bowl of a food processor and pulse once or twice to mix. Combine oil, honey and 1/2 cup warm water (between 105° and 115°F) in a small glass or measuring cup. With food processor motor running, slowly pour water mixture through feed tube; process until a ball of dough forms. Let it run until dough is smooth, about another minute. Dough should be slightly tacky but not too sticky. If necessary, add a little flour, one tablespoon at a time, and let it run for a little longer.
Oil a medium-size mixing bowl. Form dough into a ball and transfer to bowl, turning dough to coat with oil. Cover bowl with plastic wrap or a clean kitchen towel and place in a warm, draft-free spot until dough doubles in size, about 45 minutes.