Bring a large pot of heavily salted water to a boil over medium-high heat. 

Heat olive oil in a large skillet over medium heat until shimmering. Add the garlic and Calabrian Chili Peppers and cook, stirring occasionally, until the garlic is softened and slightly golden brown, about 2 minutes.

Add the tomato paste and salt. Cook, stirring often, until the tomato paste is caramelized and dark red in color, about 5 minutes. Remove the pan from the heat.

Meanwhile, add pasta to the boiling water and cook according to package directions until al dente—reserve 2 cups of pasta cooking water and set aside. Drain the pasta.

Add the pasta water to the skillet and stir to combine with the tomato paste mixture. Use a whisk to combine the mixture until smooth. Let the sauce simmer over medium-high heat for about 1-2 minutes. Reduce to low heat. Add the pasta and toss until the sauce thickens and coats the pasta completely, about 1 minute. 

Remove the pan from the heat. Add the parmigiano cheese and toss to combine. Serve with extra grated parmigiano, fresh basil leaves, and more Calabrian chili pepper paste if desired. Serve hot. Buon appetito!