Step 1Stir yeast and honey into warm water. Let stand until foamy, about 5 minutes.
Step 2Put all-purpose flour, rye flour and salt in food processor. Turn on processor and slowly pour yeast mixture through feed tube. Process dough until it just works away from sides of bowl but is still quite moist. With processor running, add 3 tablespoons olive oil through feed tube. Dough should be moist and sticky and form mass, but not ball. If dough is too sticky, add more flour; if too dry, add more water. Process until uniformly supple and elastic, 30 to 40 seconds. Or use electric mixer with paddle attachment.
Step 3Transfer dough to large plastic food bag and seal, allowing enough room for dough to double in size. Let rise in warm spot until doubled, 1 to 1 1/2 hours. When doubled, punch down. Either use dough immediately or refrigerate up to 2 days or freeze. Bring to room temperature before using.
Step 4Divide dough in 1/2. With floured hands, stretch each 1/2 into free-form circle about 10 inches wide. If dough resists reshaping, let it rest briefly and try again. Place dough on large oiled baking sheets sprinkled with cornmeal.
Step 5Heat 1/4 cup olive oil and garlic until simmering. Pour through fine sieve, pressing garlic to release flavor. Discard garlic. Brush each piece of dough with warm oil. Top with onion slices and sprinkle with red pepper flakes, salt and rosemary.
Step 6Bake at 500 degrees until lightly browned on edges and well browned on underside, about 10 minutes, reversing pans once for even browning.
Step 7Remove from oven and serve warm, cutting into quarters with kitchen shears. (Focaccia can be baked ahead and reheated wrapped in foil at 300 degrees until hot, about 12 minutes.)