To make the carrots, preheat the oven to 325°F (165°C).
Toss the carrots with a bit of olive oil and season lightly with salt; their flavor should remain sweet.
Pour just enough coffee beans into a heavy Dutch oven or ovenproof roasting pot to completely cover the bottom. Add the carrots, cover, and roast until very tender, 1 to 1 1/4 hours. Let cool in the pan; the carrots will harden as they cool. Meanwhile, to make the granola, put the oats in a medium bowl. Put the brown sugar, butter, and honey in a small saucepan over medium heat. Cook, stirring constantly, until melted and simmering, then immediately pour over the oats. Add the chicory and salt and stir until thoroughly combined.
Spread the mixture in an even layer on a rimmed baking sheet and bake, stirring occasionally, for about 25 minutes, until golden brown. Let cool completely; the granola will get crunchy as it cools. Stored in a tightly sealed container at room temperature, the granola will keep for 1 week.
To serve, remove the carrots from the pan, discarding the coffee beans. Slice the carrots on an extreme diagonal to make long slices. Spread 2 tablespoons of the crème fraîche on each serving plate and top with one-fourth of the carrot slices. Garnish each serving with a few drops of olive oil, a sprinkling of crunchy sea salt, and a few cilantro flowers. Top with small pieces of the granola and a dusting of ground coffee. Serve immediately.
Substitutions: You can use another decaffeinated coffee in place of the Blue Bottle Decaf Noir.
Note: Ground chicory root is available in natural food stores.
Andrea Reusing is the creator of the restaurant Lantern in Chapel Hill, N.C., and author of the book Cooking in the Moment: A Year of Seasonal Recipes. In this installment of The Key 3, she shares with Lynne Rossetto Kasper the techniques behind three of her favorite recipes: Turnip Soup, Overnight Braised Short Ribs and Tomato Salad.