Preheat grill to 200ºC. Place eggplant, cut-side up, on a baking sheet and sprinkle each side with salt. Let sit, 30 minutes.
Pat eggplant halves dry and brush with oil. Place each half on grill, cut-side down, and grill, 4 to 5 minutes. Flip each half and continue to grill, 2 minutes more
Remove from heat and season with salt and pepper. Drizzle eggplant with balsamic glaze and top with crumbled feta, red pepper flakes, and fresh oregano leaves. Serve.