Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente. Just before draining, reserve 1 cup of the pasta cooking water. Drain the pasta.

Meanwhile, drain the sun-dried tomato oil into a large skillet. Reserve the tomatoes. Set the skillet over medium heat. When the oil shimmers, add the garlic, oregano, crushed red pepper flakes, 1/4 cup basil, and lemon zest. Cook for 2-3 minutes, until the garlic is fragrant. Add the tomato paste and cook for another minute. Stir in 3/4 cup of the reserved pasta water and bring to a simmer. Add the sun-dried tomatoes and pasta, tossing to coat. Stir in the parmesan and a pinch of pepper. Cook until warmed through, 1-2 minutes.

Add the remaining 1/4 cup basil, toss until the herbs are wilted and the sauce coats the pasta, about 2-3 minutes, adding in more pasta water if needed to thin the sauce to your desired consistency. Remove from the heat.

Transfer the pasta to serving bowls and top each with burrata. Drizzle with olive oil, sprinkle with dill, and season with salt and pepper to taste. Enjoy immediately!