Split the leeks in half. Run the cut sides under water, fanning the layers to remove any trapped silt or sand. Cut the leek halves into 2-inch pieces. Place the leeks in a colander to dry.
Add the peaches and chiles to a mason jar with a lid. Heat the vinegar, sugar, mustard seed, cloves, 1 teaspoon salt and 1/2 cup water in a small saucepan, stirring, until the sugar is dissolved, 2 to 3 minutes. Pour the vinegar mixture into the jar with the peaches. Leave the jar uncovered to come to room temperature while you cook the leeks.
Heat 2 tablespoons of the olive oil in a heavy-bottomed skillet with a lid over medium-high heat. Add the leeks to the oil cut-side up. Cook until the leeks crisp and char, about 3 minutes. Flip, cover and cook until the leeks brown on the cut sides and begin to soften, about 5 minutes. Remove the leeks to a serving platter.
Measure 2 tablespoons of the pickling liquid into a measuring cup, jar or small bowl. Whisk in the lime juice and remaining 2 tablespoons olive oil.
Spoon a few peaches and chiles over the leeks, then pour over the dressing and sprinkle with mint. Serve immediately.