Trim and discard bottom 1 inch of broccoli rabe stems. Wash broccoli rabe with cold water, then dry with clean dish towel.
2A. For a charcoal grill Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.
2B. For a gas grill Turn all burners to high, cover, and heat grill until nhot, about 15 minutes. Turn all burners to medium-high.
Clean and oil cooking grate. Toss broccoli rabe with 2 tablespoons oil and ¼ teaspoon salt. Brush eggplant with remaining ¼ cup oil and sprinkle with remaining ¼ teaspoon salt. Arrange eggplant and lemon cut side down on grill and cook, turning as needed, until eggplant is tender and lightly browned and lemon is lightly charred, 8 to 10 minutes. Transfer eggplant and lemon to platter as they finish cooking and tent with aluminum foil.
Arrange broccoli rabe on grill and cook, turning as needed, until tender and lightly charred, 8 to 10 minutes; transfer to platter with eggplant.
Place burrata on plate and cut into 4 pieces, collecting creamy liquid. Arrange vegetables evenly on individual plates and top with burrata, pesto, tarragon, pistachios, and jalapeño, if using. Serve with grilled lemon.