Wipe the eggplant clean and slice them in half lengthwise. With the tip of a knife, score the flesh deeply in a diamond cross-hatch pattern by making two or three long cuts, cutting at a steep angle, and then rotating the eggplant to make another set of similar cuts. Press on the edges of the halves to open the cuts and sprinkle salt (1 to 1-½ tsp. total for all the halves) over the surface and into the cuts. Set aside, cut side up, for 30 min. Heat the oven to 400°F. Line a baking sheet with parchment.  Over the sink, gently squeeze the eggplant to extract the salty juice and wipe them dry with a paper towel. Brush each half thoroughly with olive oil.  Arrange each half, cut side down, on top of a sprig or two of savory on the baking sheet. Roast for 1 hour. The eggplant will collapse and the bottoms will be a deep brown caramel color. Let cool considerably before handling, at least 20 min. Gently turn the cut side up. If serving as a side dish, serve with a lemon wedge for squeezing or drizzle with vinaigrette.