By stirring the black-olive paste into the yolks, the mixture resembles a quail-egg shell.
Prepare an ice bath in a medium bowl. Place eggs in a medium pan; cover with several inches of cold water. Bring to a boil, covered. Uncover pan, reduce heat to a simmer, and cook 2 minutes, stirring gently several times. With a slotted spoon, immediately transfer cooked eggs to ice bath. Let cool completely, about 5 minutes. Peel eggs by rolling gently along work surface with palm of hand to crack shell. Hold eggs under cold running water; gently pull away membrane with shell attached. Place eggs in bowl, cover with plastic wrap, and refrigerate until ready to use.
Cut eggs in half lengthwise. Remove yolks; place in small bowl. With back of fork, mash until smooth. Stir in mayonnaise and 2 teaspoons olive paste. Season with salt and pepper. Transfer mixture to small pastry bag with 3/8-inch tip. Pipe into empty egg-white halves. Garnish tops with dollops of olive paste and a sprig of thyme each.
The eggs can be hard-cooked and peeled up to 6 hours in advance.