Hard-boil the eggs. I’m sure you have a favorite way to boil yours, but just in case, here’s mine. Put them in a large pot, so that they can sit in a single layer. (That’s important.) Add cold water to cover by an inch or two, and place the pot over medium-high heat. Bring to a boil; then immediately cover the pot and remove it from the heat. Let sit exactly 12 minutes. Drain and rinse well with cold water.

Warm the oil in a small skillet over medium-high heat. When it’s hot and runs easily around the pan, add the capers. They should sizzle. Fry, shaking the skillet occasionally, until they split open and start to crisp, about 3 minutes. They should not brown. Fish them out of the skillet, leaving the oil behind, and drain them on a paper towel.

When the eggs are fully cool, cut them in half crosswise. (I think this looks prettier – and is easier to eat – than cutting them in half lengthwise.) Carefully remove the yolks and put them in a medium bowl. Trim a tiny sliver off the rounded end of the whites, so that they will sit upright when you serve them; then set them aside. Using a fork, mash the crap out of the yolks. Add the mayonnaise, mustard, lemon juice, and salt. Mash and mix until smooth. Taste, and adjust as necessary.

Just before serving, spoon the filling into a pastry bag, and pipe the filling into the egg white halves. If you don’t have a pastry bag, spoon the filling instead. Top the filled eggs with a small dollop of basil aioli. (You will have aioli left over.) Finish with a few fried capers.