Preheat the broiler.
In a small bowl, whisk together the oil, vinegar, garlic, rosemary, 1/8 teaspoon salt, and a good pinch of pepper.
Quarter the radicchio lengthwise into wedges. Swish them around in a bowl of water and pat dry. Rub each radicchio wedge with the oil mixture and place them on a broiler pan. Broil about 6 inches from the heat source, watching closely but without turning, until slightly charred, 3 1/2 to 5 minutes.
Baste the tops of the wedges with the oil mixture and then turn them over and baste the other side. Continue to broil until browned and crisped on the surface and completely tender inside, 2 to 4 minutes. Transfer to a cutting board. Turn off the broiler and preheat the oven to 175°F (79°C).
Cut the radicchio crosswise into thin slivers. Have the melted butter, tortillas, tapenade, and goat cheese ready near the stove. Place a dry griddle or large cast-iron skillet over medium-high heat and brush the pan with some of the melted butter. Thinly spread one side of a tortilla with about one third of the tapenade. Place in the pan, tapenade side up, and immediately scatter with one third of the crumbled goat cheese, leaving a 1/4-inch border uncovered around the edge. Top with 1/3 of the slivered radicchio and place a tortilla on top; brush with a little more butter.
Press down with a spatula and cook until the goat cheese begins to liquefy and the underside is mottled brown, 3 to 5 minutes. Turn over with a wide spatula and cook until the other side is golden, about 2 minutes. Transfer to a large baking sheet and keep warm in the oven while you make the remaining 2 quesadillas in the same way.
Transfer the quesadillas to a cutting board and, using a large, sharp knife, cut each one into 6 equal wedges. Fan the wedges on a warmed platter and serve at once. Originally published August 2, 2009.