Bring a large pot of well-salted water to a boil and cook the rigatoni until al dente according to package directions; reserve 1/2 cup pasta water, then drain.
Heat the olive oil in a large skillet over medium heat. Add the minced garlic and shallot and sauté until fragrant, about 1–2 minutes.
Add the ground chicken to the skillet, season with kosher salt and black pepper, and cook, breaking it into pieces, until no longer pink, about 5–6 minutes.
Stir in 2–3 tablespoons gochujang paste until evenly combined with the chicken.
Pour in the chicken broth and scrape up any browned bits from the pan, then stir in the heavy cream. Reduce heat to low and simmer until the sauce slightly thickens, about 3–5 minutes.
Add the baby chard or spinach and cook until wilted, about 1–2 minutes.
Toss the drained rigatoni into the skillet to coat with the sauce; add reserved pasta water a little at a time to loosen the sauce if needed. Taste and adjust salt if necessary.
Serve immediately topped with sliced green onion, grated Parmesan, sesame seeds, and crushed red pepper flakes as desired.