Torn pieces of burrata are tossed into the orecchiette just after it’s cooked so the cheese melts ever so slightly into the pasta, which is laced with chunks of tender, smoky eggplant. Slideshow: More Pasta Recipes
Heat a grill pan. Rub the eggplant with 1/4 cup of the olive oil and season with salt and pepper. Grill the eggplant over moderate heat, turning, until golden and tender, about 15 minutes. Transfer to a cutting board and let cool slightly. Cut into 3/4-inch pieces and transfer to a bowl. Add the lemon juice, vinegar and 1 teaspoon of the preserved lemon rind and mix well.
In a small skillet, heat the remaining 2 tablespoons of oil. Add the garlic, oregano sprigs and chile de árbol and cook over moderately low heat, stirring, until the garlic is fragrant and starts to brown, 1 to 2 minutes. Add the mixture to the eggplant and toss to evenly coat.
In a large saucepan of salted boiling water, cook the orecchiette until al dente. Drain, reserving 1 cup of the cooking water.
Return the pasta and pasta water to the saucepan and add the eggplant and its marinade, the Calabrian chiles, the 1/2 cup of grated Romano and the remaining 1 teaspoon of preserved lemon rind. Season with salt and pepper and toss over moderately high heat until well combined and the pasta is saucy, about 2 minutes. Remove the pan from the heat and discard the oregano sprigs. Add the burrata, toss quickly and transfer to plates. Garnish with oregano leaves and chopped parsley and serve, passing grated Romano cheese at the table.
The marinated eggplant can be refrigerated overnight.
Preserved lemons, cured in lemon juice and salt, are available at specialty-food stores.