Special equipment: mortar and pestle
Pick some of the leafy fronds from the tops of the fennel and reserve. Trim and reserve the stalks for plating and for cooking the fish in the Mackerel recipe. If the outer layer of any of the bulbs is bruised or seems somewhat dried out, remove and discard a layer. Cut the bulbs in half, lengthwise, and put them on a flat surface. Cut each half into 3 equal pieces so they look somewhat like sections of an orange wedge.
Heat a large pan and add the olive oil. When the oil slightly hot, remove the pan from the heat. Arrange the fennel in a single layer in the bottom of the pan and season with the salt, and red pepper flakes. Return the pan to the heat and cook, undisturbed, until the fennel turns light brown on the first side, 3 to 5 minutes. Use a spatula or tongs to turn the fennel on its other side. When it browns on the second side, remove the pan from the heat and add the white wine. Season with salt, to taste, and return the pan to the heat. Cook over a low flame until all of the wine has reduced. How will you know? You will hear the sizzle of the fennel in the oil when the wine dries up. Insert a knife into a few of the fennel pieces. They should be tender and yielding. If not, add 1/2 cup of the water. Cook until the water evaporates. If necessary, add up to another cup (or maybe more if the fennel bulbs were unusually large), in small increments, to cook the fennel until it is tender. Taste a tiny piece for seasoning. Ideally, there should be very little liquid remaining at the end of the process. Remove the fennel from the pan and set aside to cool to room temperature.
Use a mortar and pestle to grind the walnuts. Alternatively, chop them with a large knife. In a small bowl whisk together the walnuts with the lemon juice, a pinch of sea salt, the hazelnut oil and the orange liqueur. Taste for seasoning. Set aside.
Transfer the fennel to a serving bowl and toss with some of the walnut pesto. Add the orange sections and arugula leaves with 2 tablespoons fennel stalk slices and toss to blend. Taste for seasoning. Serve immediately.