Fennel-and-Sweet-Onion Pizza with Green Olives

In the bowl of a standing mixer fitted with the dough hook, combine the yeast, honey and 1/4 cup of warm water. Let stand until foaming, about 4 minutes. Add the 2 1/2 cups of all-purpose flour and the whole-wheat flour, ale, olive oil, salt and 1/2 cup of water and mix at medium speed until a smooth dough forms, about 5 minutes. Transfer the dough to a work surface and knead for 2 minutes. Lightly oil the bowl, return the dough to it and cover with plastic wrap. Let the dough rise in a warm place until very billowy, about 1 hour. In a skillet, heat 1 tablespoon of the olive oil. Add the fennel and cook over moderate heat, turning once, until lightly browned, 5 minutes. Remove from the heat and add the wine and Pernod. Return the pan to moderate heat and cook until the liquid has evaporated, about 2 minutes. Add the chicken stock to the skillet. Cover and simmer over moderately low heat, turning the fennel once, until very tender and the liquid is nearly evaporated, about 20 minutes. Season with salt and pepper. Transfer the fennel to a cutting board and coarsely chop it. Meanwhile, in another skillet, heat the remaining 1 tablespoon of olive oil. Add the onion, cover and cook over moderate heat, stirring once or twice, until softened, about 3 minutes. Uncover and cook until the onion is caramelized, about 20 minutes. Add a few tablespoons of water to the skillet to prevent scorching. Preheat the oven to 500°. Preheat a pizza stone as close to the oven bottom as possible for 20 minutes. Punch down the dough and divide it into 4 balls. Set the balls on an oiled baking sheet and cover with oiled plastic wrap. Let stand for 15 minutes. On a floured surface, roll or stretch one dough ball to a 10-inch round. Set the round on a floured pizza peel, shaking it gently so it doesn’t stick. Brush the edge of the dough with oil. Add one-fourth of the cheese, followed by one-fourth each of the braised fennel, caramelized onion and olives. Slide the pizza onto the stone and bake until bubbling on top and the crust is deeply golden, 8 to 10 minutes. Transfer the pizza to a cutting board, sprinkle with basil and cut into wedges. Repeat with the remaining dough and toppings.