Place potato, onion, and garlic in heavy saucepan and cook on medium low until everything is soft with little color, approximately 30 minutes.
Add Dijon, stock, and cream, and simmer for 45 minutes; remove from heat and let cool at room temperature for 1 hour.
Place the mixture into a professional blender and begin to incorporate the arugula. Begin on slow speed while gradually turning higher. Continue this process until all of the arugula is incorporated; the mixture should be a smooth puree with bright green color.
Add honey wine vinegar and season with salt and pepper. Let chill completely. Poach egg in slightly acidulated water (using lemon juice, lime juice, or vinegar) until soft yolk stage. Pour 5 to 6 oz. of soup into a bowl and place warm poached egg in the center. Garnish with extra virgin olive oil and a little arugula. Serve.