Bring a large pot of salted water to a boil and add your pasta shells. Cook them until al dente, following the package timing as a guide. Drain well and set the shells aside while you work on the chicken and sauce.
Pat the chicken breasts dry, then season both sides with salt, pepper, and paprika. Heat olive oil in a large skillet over medium-high heat until it shimmers. Sear the chicken for 5 to 6 minutes per side until golden brown and fully cooked, then set it aside on a plate.
In that same skillet, melt the butter over medium heat, scraping up any browned bits left from the chicken. Add the minced garlic and sauté for about 30 seconds, just until fragrant. Be careful not to let it brown, since burnt garlic turns bitter fast.
Pour in the heavy cream and chicken broth, then let the mixture simmer for 2 to 3 minutes until it starts to thicken slightly. Stir in the shredded cheddar and grated Parmesan a little at a time until fully melted and smooth. Season with salt and pepper to taste.
Add the cooked pasta shells directly into the skillet, tossing gently so every piece gets coated in that cheesy sauce. Nestle the seared chicken breasts on top of the pasta, letting them warm through in the sauce for a minute or two. This lets all the flavors come together in one pan.
Sprinkle freshly chopped parsley over the top for a bright, fresh finish. Serve the dish immediately while it's hot and the sauce is at its creamiest. This is the moment where all that effort really pays off.