Heat a large pot over medium-high heat and add 1 tablespoon of butter. Add the cubed chicken and season with onion powder, garlic powder, paprika, salt, pepper, and chili flakes; cook, stirring occasionally, until the chicken is cooked through, about 5 minutes.
Toss the cooked chicken with 3 tablespoons of the basil pesto, then transfer the chicken to a plate and set aside.
Reduce heat to medium and add the remaining 1 tablespoon of butter and the chopped garlic to the same pot; cook until fragrant and lightly golden, about 1 minute.
Add the pasta and 2 cups of water to the pot. Increase heat to bring to a simmer and cook, stirring often, until the pasta is al dente, about 8 minutes (add a bit more water if needed to finish cooking).
Stir in the milk, cream cheese, 1/2 cup Caesar dressing, the remaining pesto, and the grated Parmesan; cook and stir until the sauce is smooth and creamy.
Return the reserved chicken to the pot, add the arugula or baby spinach, lemon juice, chopped basil, and chopped dill or parsley; toss to combine and wilt the greens slightly.
Adjust seasoning with salt and pepper if needed and serve immediately with extra grated Parmesan and basil if desired.