Step 1 Rinse the anchovies under running water to remove excess salt and then soak in water to cover in small bowl until softened, about 5 minutes. Remove the fillets, discard the skeletons and soak another 5 minutes until flexible. Cut into approximately half-inch crosswise pieces.
Step 2 Using a very sharp knife, slice the garlic crosswise as thin as you can. Reserve the tops, bottoms and odd-shaped pieces in a bowl and cover with olive oil.
Step 3Trim the tops and bottoms of the eggplants, and cut the eggplants into 1-inch-thick crosswise slices.
Step 4Working with one slice at a time, use a paring knife to cut 5 shallow slits into the eggplant -- four at the compass points and one in the center; they should be about one-half-inch deep and should not go all the way through.
Step 5Insert a garlic slice in each of the slits (if necessary, widen the slit by wiggling the paring knife in it). Insert a piece of anchovy in each slit. Finally, insert a tuft of rosemary in each slit. The fillings should be as close to flush with the surface as you can make them. Repeat with the remaining eggplant slices.
Step 6Sprinkle both sides of each eggplant slice with salt and brush generously with the garlic olive oil. Grill over a moderate fire until browned on one side, 4 to 5 minutes.
Step 7Brush with more olive oil and use a spatula to turn to the other side and brush that with oil as well. Continue cooking and brushing with oil until the eggplant is browned on both sides and tender, about 3 to 4 minutes.
Step 8Remove to a platter, brush with any remaining oil and serve.