In a small unheated skillet, combine 6 tablespoons olive oil and garlic. Then, turn heat on to low and allow mixture to sizzle for 30 seconds (do not let it brown). Transfer garlic and oil to a small bowl, add rosemary, and set aside to cool.
Grease a large bowl with olive oil. In a separate large bowl, add ¼ cup water and sugar, and stir to combine. Sprinkle yeast on top, swirl to mix, and let stand until bubbling, 5 to 10 minutes. Add remaining water, garlic-rosemary oil, honey, 2 whole eggs, 1 yolk (reserve white), both flours, and salt. Mix until a soft dough forms. Knead dough until soft and smooth, but not sticky, adding more flour if needed, 5 to 10 minutes.
Transfer dough to greased bowl, turn to coat with oil, and cover with lightly-greased plastic wrap. Top with a kitchen towel and let rise in a warm area until doubled in size, about 2 hours.
Line a large baking sheet with parchment paper. Turn dough out onto a lightly-floured surface, pat down, and cut into 3 equal pieces. Using your hands, roll each piece into a 12-inch rope and place on baking sheet. Braid dough. Cover with lightly-greased plastic wrap, top with a kitchen towel, and let dough rise in a warm spot until puffed up, about 1 hour.
Preheat oven to 375 degrees F. Whisk reserved egg white with 1 tablespoon water and brush over risen loaf. Bake for 15 minutes, then loosely cover with foil, and bake until challah is deep golden and internal temperature reaches 190 degrees F, about 5 to 10 minutes longer. Let challah cool on a rack before serving.