Lemon Confit Vinaigrette

1. Make the lemon confit: In a medium saucepan set over high heat, add the lemon halves and cover with water. Bring to a boil, reduce the heat to low and simmer very gently until the lemons are very soft, about 45 minutes. Turn off the heat and transfer the lemons to a cutting board. Once they are cool enough to handle, use a teaspoon to scrape out the fruit and pulp, discard. Use the edge of the teaspoon to gently scrape away the white pith so you are left with 4 lemon cups.
2. In a small saucepan, bring the water and sugar to a boil. Reduce the heat to medium-low, add the lemon cups and simmer until translucent, about 25 minutes. Use a slotted spoon to remove the confited lemon cups and set aside to cool. (Save the simple syrup for cocktails, iced tea or other drinks.) 3. Make the vinaigrette: To the bowl of a food processor, add 1/2 of a confited lemon rind, the anchovy fillets, egg yolks, mustard, lemon juice, balsamic vinegar, salt and pepper. Process until combined, then through the feed tube drizzle in the oil in a slow and steady stream. Cover and refrigerate until serving.