Make the pesto: in a food processor, combine the basil, nuts, garlic, 1/8 teaspoon salt, and 1/4 cup of the olive oil. Pulse until smooth. With the machine slowly running, add the remaining 1/4 cup olive oil and process until smooth. Add the cheeses, then pulse just a few times to incorporate.
Cook the eggs: In a 4-quart saucepan, add the eggs and enough cool water to cover them by 1-inch. Bring the water to a boil over high heat then immediately remove the saucepan from the heat, cover with a lid and let cool for exactly 12 minutes.
Meanwhile, fill a large bowl halfway with ice and then fill with cold water.
Using a slotted spoon, transfer the eggs to the ice bath. Let cool for 10 minutes, then peel the eggs.
Slice each egg in half lengthwise, then gently remove the yolks into a medium bowl. Set the egg whites aside. Mash the egg yolks with a fork, then add the mayonnaise, 6 tablespoons of pesto, 1/4 tsp salt, and 1/8 tsp pepper and stir into a smooth paste.
Using a spatula, scoop the filling into a pastry bag fitted with a 1/2-inch round tip. Push all of the filling to one corner of the bag and press any air out of the top. Squeezing the bag from the top, to force the filling downward, pipe the filling into the cup of each egg white so it mounds a little over the top.
Garnish each egg with 1/4 grape tomato and serve.