1. Light a grill or preheat a grill pan. Brush the cut sides of the halved eggplants with olive oil and season with salt. Grill the eggplants cut side down over moderate heat until lightly charred, about 4 minutes. Turn and continue grilling until just browned and cooked through, about 2 minutes longer. Let cool. Dice the eggplants and transfer to a bowl. Pour the vinegar over the eggplant and toss well.
2. In a small saucepan, combine the 1/4 cup plus 1 tablespoon of olive oil with the garlic, marjoram sprigs, crushed red pepper and 2 teaspoons of kosher salt. Bring the oil to a simmer, then pour it over the eggplant and toss. Let stand for 1 hour. Discard the marjoram sprigs.
3. In a pot of boiling salted water, cook the orecchiette until al dente. Drain the pasta, reserving 3/4 cup of the cooking water.
4. Add the eggplant to the pot and cook over moderate heat, stirring lightly, until hot, about 30 seconds. Add the orecchiette and the reserved cooking water and cook, tossing, for about 30 seconds. Remove the pot from the heat and stir in the pecorino and parsley.
5. Spoon the pasta into bowls. Dot the pasta with the creamy burrata filling. Garnish with the lemon zest, chile strips and marjoram leaves and serve.