Position a rack in the center of the oven and heat the oven to 425°F.
Scatter the peanuts in a pie plate or other small baking dish and toss them with 1 tsp. oil and a generous pinch of salt. Roast, shaking the pan once or twice, until they are golden brown, about 5 minutes. Set aside to cool, and then coarsely chop them. Reduce the oven temperature to 375°F.
Rinse the eggplant. Trim off their tops and then cut the eggplant in half lengthwise. In a large, shallow bowl, toss the eggplant with 2 Tbs. of the oil and the red pepper flakes. Put the eggplant cut side up on a rimmed baking sheet and sprinkle generously with salt. Roast until the eggplant is tender when pierced with a fork and the flesh is a light golden brown, 10 to 12 minutes.
Meanwhile, in a small dish, whisk the remaining 3 Tbs. oil with the lime juice, honey, and 1/4 tsp. salt. Season to taste with more salt, if necessary.
With the eggplant still on the center rack, turn the broiler on to high and broil the eggplant until well browned on top, about 5 minutes. Transfer the eggplant to a serving platter. Drizzle with the dressing. Sprinkle with mint and peanuts and serve.