1. Trim the stems and most of the dark green parts of the leeks, then split lengthwise and wash thoroughly. Dry and slice thinly.
2. Heat one to two tablespoons oil in a large skillet and sauté the leeks and garlic or shallot until the edges of the leeks are beginning to brown and caramelize, about 10 minutes.
3. Transfer the leek to the bowl of a food processor. Add another tablespoon of oil to the same skillet and sauté the mushrooms until all the liquid has evaporated and the mushrooms are nicely browned, about 10–15 minutes. Add to the food processor bowl.
4. Process until finely chopped, then add mayonnaise, salt, and pepper and process until fully combined, scraping the sides at least once. Add water, one tablespoon at a time, until your desired thickness is reached. Adjust seasoning to taste.