Heat a large pot of water to boiling, and season generously with salt. Add pasta and cook according to package directions until al dente. Drain the past and immediately cover with cold water to stop the cooking. Once pasta is cooled, drain again and set it aside.
While the pasta cooks, place the bell peppers, cut side down, and garlic on a foil lined baking sheet. Place under your oven's broiler (set to HIGH) and cook until charred and blistered, about 8 to 10 minutes. Turn the baking sheet as needed to cook all the vegetables.
Remove the garlic and peppers from the oven. Immediately place the peppers in a glass bowl and cover the bowl tightly with plastic wrap. Let the peppers sit for 5 minutes.
Carefully squeeze out of its skin and onto a cutting board. Rough chop the garlic, and sprinkle with 1/2 teaspoon salt. Using the flat/wide edge of your knife, smush the garlic and salt together to make a paste. You'll need to re-pile the garlic together and run your knife's side over it a few times. Set garlic pasta aside.
Next, zest the lemon (you should have about 1 to 1/2 teaspoons of zest) and then cut it in half. Chiffonade your basil leaves (you should have about 3 tablespoons cut basil). Set lemon and basil aside.
Remove the peppers from the bowl, and peel the skin off each pepper half. Be careful, the peppers may still be hot. Once the skin is removed, slice each pepper half into thin strips and place the strips in a medium mixing bowl. Add the garlic paste and olive oil to the bowl and stir until everything is well combine and the garlic is well distributed.
Add the lemon zest and juice the lemon into the bowl (careful not to let any seeds fall in). Stir to combine. Add the cooled pasta, mozzarella and basil to the bowl. Toss to combine.
Transfer pasta salad to a serving bowl. Cover and refrigerate until ready to serve.