Build a medium-sized fire in the barbecue, or light all three burners on the gas grill. When the coals are red and dusted with ash, spread them in a tight, single layer leaving a perimeter of grill with no coals under it; they need to emit concentrated heat. When the grill has reached 450F (235C), it’s more than ready. Turn off the middle burner.

Brush each potato with olive oil on both sides, and season generously with salt and pepper.

Place the potatoes on the grill, cut-side down and brown for 2 minutes, until they have stripes on them. Turn the potatoes and brush the cut side with oil. Cover the grill and cook the potatoes until they are tender through, which will take from 25 to 30 minutes, depending on the potato. Check them every 10 minutes, and if they are browning too quickly, turn them over again. To test for doneness, pierce the potatoes with a trussing skewer – it should slide easily through the potato.

Remove the potatoes from the barbecue, and when they are cool enough to handle, scoop out two rounds with a 1/8-th inch (standard) melon ball maker, making sure to go fairly deep into the potato without piercing the skin. Break an egg into each hole – note that the quail egg shells break easily, but the inner membrane can be tough – you can pierce it with the tip of a sharp knife, then open up the shell and release the eggs. Season the eggs with salt and pepper, replace the potatoes on the grill, cover and cook until the eggs are to your liking. At 2 minutes, the yolks and whites will be slightly runny but set; 3 minutes, the yolks will be runny; 4 minutes, the yolks will be cooked through, what we might called “hard” yet they’re still tender.

Remove the potatoes from the grill. Sprinkle each egg with a bit of piment d’Espelette. Place a slice of butter atop each potato if desired, and drizzle with a bit of the olive oil that remains, season with salt and pepper, and garnish with the tarragon leaves, leaving them whole. Serve immediately.