Salt first for less oily eggplant. Eggplant soaks up oil like a sponge, but you can reduce its ability to absorb oil by salting the cut flesh and letting it sit for 30 min. or more. Then drain, pat dry, and proceed with cooking.
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 1-1/2 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 (about 1 tsp. per half for Italian eggplant, 2 tsp. per half for globe). Arrange each half, cut side down, on top of a sprig or two of thyme 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. If using in other recipes, scoop the flesh from the skin with a spoon.