Preheat oven to 400˚F, and place the oven rack in the center of the oven. Line a baking shet with parchment paper. Place the shallots in a small bowl and cover with cold water. Set aside. Juice one of the lemons, and set aside.

Slice the ends off the two remaining lemons and discard. Slice the leons into 1/2-inch rounds, and use the tip of a sharp paring knife to lift out any seeds. Arrange the lemon slices in a single layer on the baking sheet, and brush lightly with olive oil. Flip the lemon slices over, and brush the other side lightly with oil.

Roast the lemons, flipping them over every 10 minutes, until they are very tender with just a few brown spots, 20-25 minutes. The under-sides will brown before the tops, so keep an eye on them. Don't allow the lemons to crisp. Set aside until cool enough to handle.

Transfer the lemons to the bowl of a food processor fitted with a chopping blade. If there are any juices that aren't burned and blackened (there probably won't be), add those to the food processor. Drain the shallots well, and add to the food processor. Add the honey, and pulse until coarsely chopped. Add the olive oil and half the reserved lemon juice. Pulse until the mixture is smooth and creamy, with soe chunks of lemon. Season generously with salt and pepper, and add more lemon juice or honey, just a bit at a time, to taste. Transfer to a bowl, and set aside for at least 2 hours to allow the flavors to develop. Serve at room temperature, stirring in the chopped basil or mint just before serving.

Store covered in the refrigerator for several days.