Roasted Garlic Purée: Preheat oven to 300 F. Slice off the top of the bulb, just enough to expose the cloves. Place garlic in a small baking dish, cut side down. Add enough vegetable oil to cover the garlic halfway. Roast in the oven until cloves are golden and soft, about 1 hour. Remove from oven and set aside until cool, 20 minutes. Fit a fine-mesh strainer over a bowl and press the cloves through the mesh with your hands, a bench scraper, or spoon; this is to catch any tough fiber or skin. Set aside. Makes about ¼ cup. (Note: You can roast several heads of garlic at a time and store the purée in an airtight container in the refrigerator for up to 1 month.)
Lettuce: Remove and discard tough outer leaves of the lettuce. Cut away and discard the root base, then separate leaves, cutting the larger ones in half vertically and keeping smaller ones whole. Fill the bowl of a salad spinner halfway with ice water and insert the basket. Add lettuce to the basket and submerge completely. Soak for 10 minutes to crisp up the leaves.
Sesame Croutons: In a medium skillet over medium heat, add butter and oil. Smash garlic cloves with the flat side of a knife, and add to butter and oil. Cook until bubbly, stirring occasionally, 2–3 minutes. Add breadcrumbs and toss or stir to combine. Season with freshly ground black pepper, about 8 grinds, and a pinch of salt. Continue toasting the crumbs until golden brown, 2–3 minutes. Drain on paper towels to cool.
(Note: The video shows a double serving of dressing. Use the amounts in the written recipe.)Dressing: In a blender, add mayonnaise, anchovy, fish sauce, and the Roasted Garlic Purée. (Save the remaining purée for another use.) Use a Microplane to grate the garlic clove into the blender; then add Parmigiano-Reggiano, lemon juice, water, and sugar. Season with freshly cracked black pepper (about 10 grinds) and a pinch of salt. Purée just until combined. With the motor on, drizzle in olive oil, then vegetable oil. Turn motor off, taste for seasoning, and add more salt and sugar as needed. (Makes about 1¼ cups. Store in the refrigerator in an airtight container for up to 1 week.)
Assembly: Remove ice water from salad spinner and spin the lettuce leaves twice, draining water each time, until dry. Lay out on a sheet tray lined with paper towels to further drain. Remove garlic cloves from Sesame Croutons and discard. Place half of the lettuce into a bowl. Drizzle several spoonfuls of dressing, add 1 tablespoon Parmigiano-Reggiano, and toss to combine. Repeat with remaining lettuce. Place on a serving dish, garnish with plenty of Sesame Croutons, and serve.