Combine water, honey, and yeast; let stand until foamy, 5 minutes, then stir in 1 Tbsp. oil. Pulse all-purpose flour, whole-wheat flour, and 1 tsp. salt in a food processor; add yeast mixture, and pulse until a dough forms around blade. Transfer dough to a lightly oiled bowl, turn to coat, cover with plastic wrap, and let rise in a warm place until nearly doubled, about 2 hours. Meanwhile, purée spinach, walnuts, garlic clove, and 1 Tbsp. oil in food processor; season pesto with salt and black pepper.
