Cook the chicken breast in a sous vide water bath set to 63-65˚C for one hour.
Dice the garlic clove and the anchovy together on the cutting board. Sprinkle with kosher salt and, using the side of the knife, mash them together into a paste. Scrape into a small bowl. Add the egg yolk, mustard, and lemon juice to the bowl and whisk together. While whisking, slowly drizzle in the olive oil to create a stable emulsion. Reserve in the refrigerator until ready for use.
Tear the baguette into one inch cubes. Place on a sheet pan, drizzle with EVOO, sprinkle with salt, and place in a 350˚F oven until well dried, about 10-15 minutes.
Heat a broiler on high. Cut the heart in half lengthwise and place, cut side up, on a sheet pan. Place until the broiler and broil until well charred, about 5 minutes.
Place the broiled romaine hearts on two plates. Slice the chicken breast into 1/2-inch slices and place on top of the hearts, dividing equally. Top with croutons, and drizzle the dressing. Shave some real parmigiano-reggiano over the salads.