Place cheese in freezer. In a large pot of boiling salted water, cook pasta until al dente. Reserve 1/2 cup pasta water; drain pasta and set aside.
In pasta pot, heat oil over medium-high. Add cherry tomatoes, sun-dried tomatoes, garlic, and pasta water. Cook until cherry tomatoes soften, 2 to 4 minutes.
Add pasta, chilled cheese, and chives to pot, season with salt and pepper, and toss to combine.
Chilling the mozzarella cubes in the freezer for 15 minutes helps them retain their shape when they're added to the hot pasta.