Slice the eggplants into rounds about 1 cm / 0.4 inch thick. Layer them in a colander, sprinkling coarse salt between layers, and let sit for 30 to 45 minutes until beads of moisture appear on the surface.
Rinse the salt off briefly and pat the rounds dry with paper towels. This keeps the eggplant from turning bitter and stops the oil from spitting.
Pour the sunflower oil into a 26 cm / 10-inch skillet or deep pot to a depth of about 2 cm / 0.8 inch. Heat to 170 C / 340 F.
Fry the eggplant in batches of 5 to 6 slices so the pan isn't crowded and the oil temperature stays steady. Cook for 2 to 3 minutes per side until deep golden brown and tender at the center.
Lift each batch out with a slotted spoon onto a paper towel-lined tray and sprinkle lightly with salt while still hot.
In a small bowl, stir together the white wine vinegar, water, sliced garlic, chili flakes, and a pinch of salt.
Arrange a layer of fried eggplant in a shallow dish, scatter over some of the torn mint, and spoon over some of the vinegar mixture. Repeat layering until all the eggplant, mint, and marinade are used.
Drizzle the extra virgin olive oil over the top. Cover and let marinate at room temperature for at least 2 hours, or in the fridge overnight for deeper flavor.
Bring back to room temperature before serving, spooning some of the marinade over each portion.