Place eggs in a large saucepan and fill with cold water until the eggs are covered by about 2 inches. Cover, and place over high heat. When water boils, turn off the heat. Remove pan from hot burner (if you have an electric cooktop). Set a kitchen timer (or watch the clock) for 14 minutes. Meanwhile, fill a large bowl with cold water and ice. After the eggs are cooked, carefully transfer them from the hot water to the ice water. Let eggs cool.
When cool, lightly tap eggs on the countertop to crack shells, and peel. Cut eggs in half lengthwise with a sharp knife. Gently pop the yolks out of each white half, placing yolks into bowls and setting whites aside. Discard 1 whole white. Place whites on a plate, cover with plastic wrap, and set aside.
Place yolks, mayo, basil, Dijon, and lemon zest into a mini food processor. Process until basil is finely chopped and mixture is smooth and creamy. (Alternately, finely chop the basil and mash everything with a fork until smooth.) Place filling in a medium bowl. Cover bowl with plastic wrap, and refrigerate for 1 hour.
To serve, spoon filling into the white halves. To be fancier, spoon filling into a plastic zip-top bag, then cut a hole in one corner. Use it as you would a piping bag and pipe the filling into the whites. Sprinkle tops with big flakey salt. Serve cold or at room temperature.