For the short ribs: Heat the oil in a medium Dutch oven over medium-high heat. Sprinkle the short ribs evenly with the salt and add them to the pan. Sear well on all sides. This will take about 15 minutes total, so be patient; the meat should have a deep golden brown color and release evenly from the pan before flipping. Remove the seared short ribs to a plate.
To the hot pan, add the carrot, celery, onion, thyme, bay leaf and pepper flakes. Cook, stirring often with a wooden spoon, until very fragrant and beginning to soften, about 4 minutes. Stir in the tomato paste and caramelize for 2 minutes, stirring constantly. Add the red wine and simmer for 5 minutes to reduce slightly. Stir in the beef broth and add the short ribs back to the pan. Bring to a simmer and reduce the heat to low. Cover the pan and simmer until the meat can be pulled apart with a fork, 2 1/2 to 3 hours. Set aside to cool slightly.
Place the short ribs in a medium bowl. Using two forks, shred the meat into bite-size pieces. Add about a 1/2 cup of the braising liquid to the meat and toss to moisten.
For the remoulade: In a medium bowl, combine the eggs, cornichons, mayonnaise, capers, Dijon, tarragon, salt and pepper. Fold together with a rubber spatula.
For the crostini: Spread a small amount of the remoulade on each of the crostini. Top with a spoonful of the shredded meat and another small dollop of the remoulade. Garnish with a slice of cornichon.
To hard-boil the eggs, place the eggs in a small saucepan and cover with water. Bring to a boil over medium-high heat, then cover the pan and turn off the heat. Allow the eggs to sit for 9 minutes. Drain and shock in ice water. Now they are ready to use. To make the crostini, brush sixteen 1/4-inch-thick baguette slices with extra-virgin olive oil, and sprinkle with kosher salt. Bake in a 400-degree oven for about 5 minutes, rotating halfway through.
Recipe courtesy of Giada De Laurentiis