Step 1Stir yeast and honey into water; let stand until foamy, about 5 minutes.
Step 2Place bread and whole-wheat flours and salt in food processor fitted with metal blade. Pulse to mix. With machine running, slowly pour yeast mixture through feed tube, then add oil. Process until dough cleans sides of work bowl. Adjust consistency of dough as needed: If it is too dry, add a little more water; if it is too sticky, add a little more flour. When dough is right consistency--moist but not sticking to work bowl--process it until supple and elastic, about 30 seconds more.
Step 3Transfer dough to large resealable plastic food bag, squeeze out air and seal at top, leaving room for dough to expand. Let dough rise in warm place until doubled in volume, about 1 hour. Press dough through bag to deflate. Use immediately or refrigerate overnight in bag. (Dough can be frozen. Bring to room temperature before using.)
Step 4Makes dough for 2 (12x10-inch) flat breads.