1. Preheat oven to 400˚
2. Trim the roots and any dark wilted tops of the leeks; pull off any damaged outer leaves. Halve the leeks lengthwise. Rinse the leek halves thoroughly in cold water (making sure to get the hidden grit at the spot where the light-colored portion starts to turn a darker green). Pat them dry.
3. Heat a large skillet over medium-high heat and add the roasted garlic oil. Working in batches, add the leeks, cut side down, and sear for 3-4 minutes. Season the leeks with salt and pepper, then turn to cook on the other side for 1 minute. Transfer the leeks to a dutch oven or a baking dish with a tight-fitting lid.
4. Add the shallots to the skillet and cook over medium-high heat until aromatic, 2 to 3 minutes. Add ¼ cup water, the broth, tomatoes, olives, thyme, lemon juice, and honey. Bring the mixture to a boil, then remove from the heat and pour it over the leeks. Be careful not to completely cover the leeks with the liquid.
5. Cover the leeks, transfer to the oven, and braise until the leeks are fork-tender, about 35 minutes. Gently stir in the basil and serve hot.