First, make the homemade mayo. Wrap a large bowl in a cold, wet towel to help it emulsify. Combine the lemon zest, lemon juice, egg, Dijon, basil and salt in the bowl and whisk them together well. Continue to whisk while you slowly pour in the canola oil. Keep whisking until it is a creamy, pale mayonnaise! Set the bowl aside while you make the turkey burgers. There will be lots leftover and this stuff will keep for days in a sealed container to put on sandwiches and other yummy things!
For the burgers, combine the basil, egg, lemon zest, Dijon, salt and pepper in a bowl. Stir it all together. Add in the ground turkey and mix it thoroughly with your clean hands. Then pre-heat an electric griddle to 350 or heat up a grill pan or griddle pan over medium high heat. Divide the burger mixture equally into quarters and form each of them into patties. Make a little thumb indentation in the middle of each of them so that they are even when they puff up as they cook.
Grease the griddle with the butter and cook the patties for 5-6 minutes on the first side. Flip them over to be close together and top each of them with the slice of mozzarella. Place a large stainless steel bowl over the patties on the griddle like a dome to steam them and really melt the cheese while they cook for another 5-6 minutes. While they cook, toast the buns. When they're done place each burger on a bun, top them with a spoonful of the mayo and a small handful of the arugula. Serve immediately with sides and enjoy!