Remove grate and prepare a charcoal grill for medium heat (coals should be covered with ash and glowing red with no black remaining). Place eggplants directly on coals and cook, turning occasionally, until skins are completely charred and flesh is collapsed, 15–20 minutes. Transfer to a rimmed baking sheet and let cool. (Or, you can grill on the grate of a gas or charcoal grill over medium-high heat, turning occasionally, 20–25 minutes.)
Split eggplants lengthwise and scoop flesh from skin into a medium bowl (it’s okay if bits of charred skin get in there too). Mash eggplant with a fork to break up, then mix in garlic, yogurt, chopped mint, red pepper flakes, and 2 Tbsp. oil; season with salt.
To serve, drizzle dip with more oil and top with mint leaves.
Dip can be made 1 day ahead. Cover and chill.