Heat 3 tablespoons of olive oil and the butter in a medium (11-inch) pot or Dutch oven, such as Le Creuset, over medium-low heat. Add the fennel and onions and sauté for 15 minutes, stirring occasionally, until they begin to brown.
Add the potatoes, chicken stock, 1 tablespoon salt, and 1 teaspoon pepper and bring to a boil. Lower the heat, cover, and simmer for 30 minutes, until the potatoes are very tender.
In batches, purée the soup in a food processor fitted with the steel blade until smooth. Pour the soup back into the pot, stir in the Pernod and half-and-half, and taste for seasonings. Reheat over medium-low heat.
Place the garnishes in the middle of large, shallow soup bowls and pour the hot soup around them. Drizzle with olive oil, garnish with the reserved fennel fronds, and serve hot.
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