Preheat the oven to 375 degrees F.
For the Caesar dressing: Beat the egg yolks in a large mixing bowl. Add the lemon juice, Worcestershire sauce, anchovy and garlic and whisk to combine. Slowly drizzle in the olive oil until the mixture emulsifies. Season with salt and pepper. Cover and refrigerate until ready for use.
For the croutons: Cut the baguette on an angle to create thin, oval slices about 1/4-inch thick. On a large baking sheet, arrange the slices evenly. Brush the bread with olive oil and top with the Parmesan and thyme. Season with salt and pepper. Bake until the Parmesan is melted and the croutons are golden brown around the edges, 5 to 7 minutes. Remove from the oven and allow to cool. Store in an airtight container if not using immediately.
For the salad: Core the romaine and trim the top 1/2 inch off the leaves. Fan the leaves out on a serving plate (or chop into 1/2-inch pieces if you prefer). Sprinkle the Parmesan over the lettuce and drizzle the dressing on top. Top with 2 to 3 croutons per plate and serve.
To coddle eggs, place the eggs place into very warm water to warm the egg through, 3 to 4 minutes, then separate and reserve the yolks.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.