FOR THE SALAD: Cut the stalks and foliage off from the fennel bulb and set aside. Halve the fennel bulb, then cut out the tougher stem from the center of each fennel bulb half. Very thinly slice the fennel and transfer to a large salad bowl.
SEGMENT THE GRAPEFRUITS: First, peel the fruit with a knife. Cut off both the top and bottom of the grapefruit. Stand the fruit on either of the flat ends. Run a very sharp paring knife along the fruit from top to bottom, removing the skin and pith, leaving the flesh bare. Rotate the grapefruit and repeat.
To collect segments, hold the peeled grapefruit in one hand, positioning it over a bowl to catch dripping juices. Run a sharp knife alongside the membranes, leaving entirely bare pieces of fruit behind. Once you've collected all the segments from the grapefruit, squeeze the remaining membranes and bits of flesh to collect all of the remaining juice. (Save this grapefruit juice to use in the dressing.)
Transfer the grapefruit segments to the salad bowl with the fennel. Add the greens.
TO MAKE THE DRESSING: In a screw-top jar, add all the dressing ingredients. Close the jar and shake vigorously.
SERVING: Drizzle about 3 tbsp (45 ml) of the dressing over the grapefruit fennel salad. Gently toss to coat. If desired, season with fleur de sel or sea salt flakes and additional freshly ground black pepper. Drizzle a bit more dressing over the salad. Generously garnish with some of the saved fennel foliage. Serve immediately.
MAKE IT VEGAN: Omit the Parmigiano-Reggiano and substitute a sharp vegan cheese, if desired.
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