Bring a large pot of salted water to a rolling boil and cook your pasta until it's just al dente, checking a minute or two before the package time. Reserve about half a cup of the starchy pasta water before draining. It's a secret weapon for loosening the sauce later. Set the pasta aside while you work on the chicken and sauce.
Heat olive oil in a large skillet over medium-high heat and add the diced chicken in a single layer. Let it sit undisturbed for a few minutes so it actually browns instead of steaming, then flip and cook the other side. Once it's cooked through and golden, remove it to a plate and set it aside.
In the same skillet, add the minced garlic and sauté for about a minute until fragrant, then stir in the roasted Hatch green chiles. Pour in the heavy cream and chicken broth, stirring well, and bring the mixture to a gentle simmer over medium-low heat. Keeping the heat gentle here matters, since cream can curdle if it boils too hard.
Sprinkle the freshly grated Parmesan into the simmering sauce a little at a time, stirring continuously until it fully melts and the sauce thickens slightly. This slow addition keeps the cheese from clumping or turning grainy. You'll notice the sauce turning glossy and slightly green-flecked from the chiles.
Add the cooked chicken back into the skillet, stirring to coat every piece in that spicy cream sauce. Toss in the baby spinach and lime juice, stirring until the spinach wilts down, which only takes two to three minutes. Taste and adjust with salt, pepper, and cumin if you're using it.
Add your drained pasta directly into the skillet with the sauce, chicken, and spinach, tossing everything together with tongs. If the sauce feels too thick, splash in a bit of that reserved pasta water to loosen it up. Serve it hot, straight from the skillet, while the sauce is still glossy and clinging to every noodle.