How to Make It
Heat a large skillet over medium-high until very hot, 4 to 5 minutes. Toss together Broccolini and 3 tablespoons oil in a large bowl until coated. Add half of Broccolini to skillet, and spread in an even layer. Cook until slightly charred and crisp-tender, 2 to 3 minutes per side. Transfer to a large bowl, and repeat with remaining Broccolini. Toss Broccolini with 3/4 teaspoon salt.
Carefully wipe skillet clean, and return to heat over medium-high. Add 2 tablespoons oil; swirl to coat. Add half of the beans; spread in an even layer. Cook, undisturbed, until outer skins of beans begin to peel back, crisp, and brown slightly, about 1 minute and 30 seconds. Add chile and garlic, stir, and cook, undisturbed, until fragrant, 15 to 30 seconds. Transfer mixture to bowl with Broccolini. Repeat procedure with remaining 2 tablespoons oil and remaining beans.
Gently toss Broccolini and beans with parsley and remaining 1/4 teaspoon salt. Arrange burrata in center of serving bowl, and surround with Broccolini and beans mixture. Sprinkle with pecorino Romano and black pepper. Drizzle with oil, and serve with lemon wedges.