For the vinaigrette: Place the mustard and vinegar into a bowl or standing mixer fitted with a whisk attachment. Whisk by hand or at a slow speed using a mixer add the oil for a nice thick and creamy sauce. Season with salt and pepper, then it's ready to use.
For the chicken: Preheat the oven to 215 degrees F. Mix equal parts salt and pepper, and then sprinkle over all sides of the chicken. Rest the chicken in the fridge for 30 minutes.
Place the chicken on a raised rack so it doesn't sit in its own fat. Place in the oven and cook until an internal thermometer stuck in the thigh meat hits 165 degrees F, 4 hours.
The leg bone should pull out very easy. Once the chicken is done, brush the vinaigrette onto the chicken. Rest the meat for 5 minutes before eating to ensure juicy product.
While the chicken rests, pan-fry the eggs, season with salt and pepper. Using the same mustard dressing, lightly toss the arugula and serve the chicken with the salad and fried eggs.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.