Cut the squash into 3" wide lengths. Use a vegetable peeler to remove the peel. Cut each piece crosswise into slices that are a little over a quarter-inch thick.

Place the olive oil and garlic cloves in a medium sauté pan. Sauté the garlic over low heat until it turns a golden brown. Discard the garlic. Raise the heat to medium-low and fry the squash slices in the oil on both sides until they blister and turn golden brown. Cook the squash in batches, one layer at a time and without crowding the pan. Use a thin wooden skewer to check for tenderness. The squash slices should be tender but hold their shape. As each batch is cooked, transfer the slices to a platter. When all the squash is cooked, sprinkle the chopped mint over the top.

Add the vinegar, sugar, salt, and pepper to the remaining olive oil in the pan. Stir well and raise the heat to high. Let the mixture boil until it thickens slightly. Pour the hot liquid over the squash slices, then let cool. Marinate for at least several hours before serving. Lasts for several days, refrigerated.