Step 1

Preheat oven to 350°F. Cut lemon in half crosswise; thinly slice 1 lemon half. Reserve remaining lemon half for another use.

Step 2

Peel away and discard outer layer of each leek, and trim the hairy roots, leaving root ends attached. Place 1 leek on a cutting board with white end of the leek pointing away from you. To remove dark green part of leek, cut it at an angle, then roll the leek 90 degrees to repeat the same cut. Continue rolling and cutting in this manner until the tapered pale green end of leek resembles a sharpened pencil. Discard dark green leek trimmings. Cut leek lengthwise into quarters. Rinse to remove fine grit between each layer, and pat dry. Repeat process with remaining leek.

Step 3

Place leeks in a 3-quart baking dish. Drizzle evenly with 1/4 cup of the oil; sprinkle evenly with salt and pepper to taste; toss to coat. Pour in chicken stock. Drizzle evenly with remaining 1/2 cup oil. Arrange lemon slices over leeks, and scatter with thyme sprigs. Cover baking dish tightly with aluminum foil. Bake in preheated oven until leeks are very tender, about 45 minutes. Remove from oven, and increase oven temperature to 400ºF. Remove and discard foil from baking dish. Remove and discard lemon slices and thyme sprigs. Return leeks to oven, and bake at 400°F until leeks are glazed and golden brown and liquid has evaporated, 1 hour to 1 hour, 15 minutes. Set aside at room temperature until ready to use. Braised Leeks may be stored in an airtight container in refrigerator up to 5 days.

Step 4

Whisk together shallot, lemon juice, vinegar, salt, and pepper in a small bowl. Let stand 10 minutes. Gradually add oil in a slow, steady stream, whisking constantly until emulsified. Whisk in Parmesan cheese, garlic, and lemon zest. Store in an airtight container in refrigerator up to 5 days.

Step 5

Whisk together vinegar and lemon juice in a small bowl. Whisk together canola oil and olive oil in a small measuring cup. Place egg yolk in a medium bowl. Wet a clean kitchen towel, and drape over a medium saucepan to cover; place bowl with egg yolk on top of towel. This will help hold the bowl in place. Whisk egg yolk until pale yellow, 2 to 3 minutes. Add salt and garlic; whisk to combine.

Step 6

Add oil mixture 1 drop at a time, whisking constantly, until oil and egg are emulsified; you will have added 2 or 3 tablespoons of oil mixture. Add 1 tablespoon of the lemon juice mixture, and whisk to combine. Drizzle remaining oil mixture in a slow, steady stream, whisking constantly and stopping when mayonnaise thickens to add remaining lemon juice mixture, until you've added all the oil. Whisk in remaining lemon juice mixture, if there is any. Aïoli may be stored in an airtight container in refrigerator up to 3 days.

Step 7

Rinse anchovies under cold water to remove salt. Place anchovies in a small bowl, and cover with cold water. Let soak until anchovy flesh softens slightly, 20 to 30 minutes. Drain and gently pat anchovies dry. Using a paring knife, cut each anchovy lengthwise into 2 fillets. Gently lift out spines and discard. Stir together anchovy fillets, oil, parsley, and lemon zest in a small bowl. Let stand at room temperature 1 hour.

Step 8

Meanwhile, bring 8 cups water to a boil in a large saucepan over high. Add salt. (The salt helps the egg whites solidify quickly if a shell cracks). Gently add eggs; simmer, undisturbed, for 5 minutes. Meanwhile, fill a medium bowl with ice water; set aside. Remove pan from heat; let eggs stand in water 5 minutes. Remove 1 "tester" egg, and crack it open to see if it's cooked to perfection — white and the outer rim of the yolk should be cooked through, while the innermost portion of yolk should be bright yellow and slightly wet looking. If the yolk is runny and wet looking, let eggs stand in hot water up to 2 additional minutes. Transfer eggs to prepared ice water; let cool completely, about 10 minutes. Peel eggs, and cut into thin slices. Set aside until ready to use.

Step 9

Preheat grill to medium-high (400°F to 450°F). Arrange bread on oiled grates; grill, uncovered, until toasted, 2 to 3 minutes per side. Remove from grill. Rub garlic clove evenly over 1 side of each bread slice.

Step 10

Spread about 2 tablespoons Aïoli over the garlic side of each bread slice. Cut each bread slice in half diagonally, and divide evenly among 6 plates. Cut 4 Braised Leek pieces diagonally into thirds; arrange evenly on top of bread pieces, fanning them out slightly. Reserve remaining leeks for another use. Arrange egg slices evenly on top of leeks. Top each bread piece with 2 anchovy fillets, and drizzle with some of the anchovy marinade. Sprinkle evenly with chives.

Step 11

Toss together lettuce leaves and 3/4 cup Garlic-Parmesan Vinaigrette in a medium bowl. Massage each leaf with vinaigrette to ensure every leaf is dressed evenly. Divide lettuce evenly among plates, and fan out next to the crostini.