Fill a large pot with water, salt it generously, and bring to a rolling boil. Cook the rigatoni according to the package instructions until al dente, about 11 minutes. Scoop out 1/2 to 1 cup of the pasta water, then drain the pasta.

While the pasta cooks, place the chicken in a large mixing bowl. Season with the oregano, kosher salt, and black pepper. Toss to coat evenly.

Warm 3 tablespoons of the olive oil in a large, deep-sided skillet over medium-high heat. Add the chicken and cook for 7 to 10 minutes, stirring occasionally, until browned on all sides and cooked through. Transfer the chicken to a plate using a slotted spoon.

Add the remaining 3 tablespoons oil to the same skillet. When shimmering, add the chopped basil and cook, stirring, for about 1 minute until fragrant. Stir in the garlic, sun-dried tomatoes, and crushed red pepper flakes; cook for 1 to 2 minutes more.

Add the tomato paste and stir to break it up. Pour in the white wine and let it reduce for 1 to 2 minutes. Pour in the heavy cream, scrape up any browned bits from the pan bottom, and stir until the sauce is smooth.

Return the chicken to the skillet along with the cooked pasta, spinach, 1/2 cup of the reserved pasta water, and the grated Parmigiano-Reggiano. Toss well, adding more pasta water if needed to create a silky sauce. Taste and adjust seasoning. Serve immediately in bowls, topped with extra grated Parmesan, basil, and a pinch of flaky sea salt.