Step 1Heat oven to 400 degrees. Remove the greens from the beets, coarsely chop the greens and reserve.
Step 2Place the beets on a double layer of foil and drizzle over 3 tablespoons oil. Sprinkle over ¼ teaspoon salt and add the rosemary. Seal the foil around the beets, making a pouch, place on a baking sheet and bake until the beets are tender, about an hour (a knife should pierce the beets easily). Set aside until cool enough to handle, then peel and thinly slice.
Step 3In a skillet heated over medium-high heat until hot, add 2 tablespoons oil. Stir in the garlic, cooking for a minute or so until it begins to color, then add the chopped greens. Cook, stirring frequently, until the greens are wilted and the stems are tender, 10 to 15 minutes. Remove from heat and set aside.
Step 4Heat a cast iron skillet or griddle over medium-high heat until hot and coat with a thin layer of olive oil. Place a couple tortillas in the pan and top with goat cheese, beet greens and sliced beets. Top with tortillas, flattening each to spread the filling evenly.
Step 5Cook the quesadillas until the cheese is melted and the tortillas are golden, carefully flipping to cook the filling and tortillas evenly. Repeat with the remaining tortillas and filling. Halve the quesadillas and serve while warm, with hot sauce on the side.