For the aioli: Combine the mayonnaise, basil, lemon zest and juice and garlic in a blender and blend until almost smooth. Season with salt. Transfer to a bowl and refrigerate until ready to serve.
For the carrot fries: Preheat the oven to 425 degrees F. Fit a large baking pan with a wire rack and spray the rack with cooking spray.
Mix together the panko, Parmesan, parsley, cayenne and 1/2 teaspoon salt in a wide shallow bowl. Whisk the milk and eggs together in another shallow bowl.
Toss the carrots with the flour. Working with a few carrots at a time, shake off the excess flour, then dip them in the egg. Let the excess egg drip off, then toss them in the panko mixture, patting them so the crumbs adhere. Arrange them on the wire rack in a single layer.
Spray the coated carrots thoroughly with cooking spray, then bake until golden brown and crisp-tender, 10 to 15 minutes.
Serve the carrot fries with the aioli for dipping.