Bring a large pot of salted water to a rolling boil. Add the chickpea pasta and cook until just shy of al dente according to the package (usually a minute or two less than recommended). Reserve 1 cup of the starchy cooking water, then drain the pasta and set aside.
While the pasta cooks, heat the olive oil in a large wide skillet over medium-high heat. Add the diced onion and bell pepper and sauté until softened and starting to color, about 4 minutes.
Push the vegetables to one side of the pan, add the ground beef to the empty space and let it sear undisturbed for 1–2 minutes to develop color. Break the beef up with a spatula and continue cooking until no pink remains and liquid has mostly evaporated, about 5 minutes.
Stir the minced garlic into the beef and vegetables and cook 30 seconds until fragrant. Sprinkle in the oregano, paprika, 1/2 tsp of the salt and the black pepper; stir to coat and toast the spices briefly.
Mix the tomato paste into the pan and cook for 1 minute to deepen its flavor, then pour in the tomato sauce. Add a 1/4 cup of the reserved pasta water, scrape up any browned bits, and bring the sauce to a gentle simmer. Taste and add the remaining 1/4 tsp salt if needed. Let the sauce simmer, uncovered, for 5–7 minutes to thicken slightly.
Turn the heat to low. Remove the skillet briefly from heat and whisk in the cream cheese and sour cream until smooth and silky—doing this off direct heat helps prevent curdling. If the sauce seems too thick, add up to 1/2 cup more reserved pasta water a little at a time until you reach a creamy consistency.
Return the drained pasta to the skillet and toss gently to coat evenly, heating together for 1–2 minutes so the pasta finishes cooking in the sauce. Sprinkle in the grated Parmesan and torn basil, fold through, and let the pasta rest in the warm pan for 2 minutes—this helps the sauce cling to the noodles.
Serve immediately in warmed bowls and adjust seasoning with extra salt or black pepper if needed. For a brighter finish, add an extra torn basil leaf or a small grating of Parmesan to each serving.