Bring enough water to a boil to cook the pasta. When the water is boiling, pour in the pasta and cook according to package directions.
While pasta is cooking, in a large sauté pan, heat olive oil over medium heat.
Add whole garlic cloves to oil and sauté until all sides are golden and the garlic is soft on the inside. (About 4-5 minutes). Reduce heat if garlic is browning too quickly.
Turn heat in pan up to medium high and add in whole tomatoes (be careful of splattering if the tomatoes are wet) and gently sauté the tomatoes in the oil until they start to lose their shape and the skins burst.
Remove pan from the heat. Using a potato masher or large fork, mash each tomato and garlic clove into the olive oil. This will give you a very rustic sauce with pieces of tomato and garlic in the oil. (Alternately, you can pour the mixture into the deep sided bowl and blend with an immersion blender for a smoother, more emulsified sauce.)
Drain cooked pasta and add it into the sauce along with the asparagus, chicken, fresh basil and Parmesan cheese. Toss well to combine. Adjust seasoning with salt and pepper.
Serve pasta with additional grated Parmesan.
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