Make the socca: In a large bowl, whisk all the ingredients together until a smooth batter forms. Cover in plastic wrap and let sit for 1 hour.
Meanwhile, make the chickpea pesto: Bring a small pot of water to a boil and prepare an ice bath. Season the water with salt. Add the fresh chickpeas and cook until tender, 2 minutes. Using a slotted spoon, transfer the chickpeas to the ice bath. Once cool, drain the chickpeas and remove their thin shells around the green centers. Transfer the chickpeas to a small food processor with the remaining pesto ingredients. Pulse until a smooth paste comes together, then season with salt.
Assemble the socca pizza: Preheat the broiler with the rack 6 inches away from the heat source. In a 10-inch ovenproof, nonstick pan, heat the olive oil over medium-high heat. Pour the batter into the pan, swirling to cover the entire surface, then transfer the pan to the oven. Bake until the socca sets, 5 to 6 minutes.
Remove the pan from the oven. Spread the chickpea pesto over the socca and dot with the goat cheese. Return to the oven and broil until the cheese is slightly golden, 4 minutes more. Transfer the socca to a board and top with the sliced serrano, chopped pistachios, mâche and chervil. Drizzle with olive oil and garnish with flaky sea salt, then serve.