Heat oil in a large skillet over medium. Add crushed red pepper and garlic; cook until fragrant, about 2 minutes. Add eggplant, and cook, stirring occasionally, until browned, about 20 minutes.
Meanwhile, cook pasta in boiling salted water according to package directions for al dente. Drain pasta, reserving 1 cup cooking water. Place cooked pasta in a serving bowl; add eggplant mixture. Slowly add reserved cooking water, tossing to coat. Tear fresh Burrata into pieces over the bowl (to catch any cream from the cheese), and add torn fresh mint, lemon zest, and lemon juice. Toss to combine. Add salt to taste, if desired. Top servings with additional mint.