For the burgers: add all ingredients to a bowl, then mix it together with your hands. Once everything is combined, form the mixture into burgers. Heat your grill as you normally would, OR heat a skillet with 2 tbsp olive oil, then add the burgers and cook until the meat is done to your liking.
For the whipped feta: add the basil, arugula, garlic, and olive oil to a food processor and pulse until everything is combined. Then, add the feta, cream cheese, and milk and pulse until the pesto and feta are mixed together. You may need to pour everything into a bowl and stir by hand a bit if your processor won't completely mix it.
To assemble: place burgers on a bun, then top them off with a spoonful of pesto whipped feta.