Place eggs in a single layer in a saucepan and cover with cold water by about an inch. Bring to a gentle boil over medium-high heat. When the water reaches a rolling boil, cover the pan and remove it from heat; let sit for 10 minutes.

Drain the hot water and transfer the eggs to an ice bath (or run under cold water) until cooled. This stops cooking and makes peeling easier. Peel each egg carefully — the shells will slide off when they’re properly cooled.

Halve the eggs lengthwise and gently scoop the yolks into a medium bowl. Arrange the empty whites on a serving platter and set aside.

Mash the yolks with a fork until smooth. Add mayonnaise, olive oil, lemon juice, minced garlic, chopped basil, capers (if using), salt, and pepper. Stir until the mixture is creamy and aromatic. Taste and adjust seasoning — a splash more lemon brightens everything.

Spoon or pipe the yolk mixture back into the egg white hollows. For a lovely presentation, use a piping bag fitted with a star tip, or simply use a teaspoon to mound the filling.

Sprinkle each egg with a tiny pinch of red pepper flakes, extra basil, and a halved cherry tomato if you like color and freshness. Chill for 15–20 minutes to let flavors marry, or serve immediately.