Step 1To prepare the fennel, cut off the stalks, reserving enough fronds to make 1 tablespoon when chopped. Cut the bulbs in quarters lengthwise, exposing the triangular core at the base. Cut out the core. Using a mandoline or sharp knife, shave the fennel very thin into a large bowl and reserve. Chop the reserved fronds.
Step 2In a small jar with a tightly fitting lid, combine the mustard, lemon juice, vinegar, olive oil, one-half teaspoon salt, chives and the chopped fennel fronds. Shake well to mix and reserve.
Step 3When almost ready to serve, season the flesh side of the salmon fillets with plenty of salt and a couple of grindings of pepper. Brush the grill and the skin side of the salmon lightly with oil. Grill skin side down over a medium-hot fire until you see the cooked color turn about one-third of the way up each fillet, 5 to 7 minutes. Lightly brush the flesh side with oil and carefully turn the fillets. Grill 3 to 5 minutes more on the flesh side.
Step 4While the salmon is cooking, vigorously shake the dressing again to combine the ingredients and pour about two-thirds of it over the shaved fennel. Toss well; the dressing should just lightly nap the fennel. Add more dressing if necessary and transfer the slaw to a serving bowl.
Step 5Carefully transfer the salmon from the grill to a platter and serve.