1. Make the butter at least 2 hours prior to preparing the chicken. Place all the ingredients for the Dijon herb butter in a stand mixer fitted with the paddle attachment, or use a handheld electric mixer, and beat on high until well combined. Place a piece of plastic wrap on the counter, then transfer the butter mixture on top of the wrap. Shape the butter into a log about 6″ x 2″ then wrap tightly in the plastic wrap. Place the log in the refrigerator and allow to chill until firm, 25-30 minutes, before slicing. Once firm, slice the log into 6 slices. Store in the refrigerator for up to 1 week. 
2. Preheat the oven to 400F. Rinse the chicken thoroughly and pat it dry all over with paper towel to remove any excess moisture. Place the chicken in a 9″ x 13″ roating pan and tuck the wings under the body. Generously season the inside of the cavity with salt and pepper, then place the lemon quarters inside. Using the back of a spoon, gently lift the breast skin of the bird and place two rounds of the Dijon herb butter under the skin atop each breast (using 4 rounds of butter in total). One the butter is in place, use some kitchen twine to tie the legs together in a simple bow. Rub the outside of the chicken with olive oil and generously season with more salt and pepper
3. Place the chicken in the oven and roast for 15-20 minutes per pound (60-80 minutes for a 4lb bird) or until the juices run clear when you make a little cut between the thigh and leg. Baste the bird with juices every 15 minutes until done. Remove from the oven and place the last 2 pieces of Dijon butter atop the chicken, then tent with foil and allow to rest for about 15 minutes. 
4. Make the dressing. Place all the ingredients for the dressing in a small liquid measuring cup and whisk to fully combine. Spread the arugula across a serving platter and gently toss with the dressing. Place the warm chicken atop the arugula and serve!

Leftover chicken and dressing can be stored in the fridge, in an airtight container, for 2-3 days.