Heat the olive oil in a large, wide, heavy-bottomed pot over medium heat.

Add the onions and a generous pinch of kosher salt. Cook, stirring occasionally, until very soft and translucent, 8 to 10 minutes.

Add the garlic and Calabrian chiles. Cook about 2 minutes, stirring frequently, until fragrant.

Increase the heat to medium-high and add the tomato paste. Stir until fully combined with the onions, garlic and chiles.

Add the butter and another generous pinch of salt. Stir until the butter has completely melted into the tomato paste.

Cook the tomato paste, stirring and scraping every minute or two, until it is deep brick red, intensely caramelized, glossy and jammy. Let the paste make contact with the bottom of the pot so browned bits develop, then scrape them up. If the pot becomes truly dry, add another small drizzle of olive oil. Because this batch is much larger than a single recipe, go by color and texture rather than a set time.

Pour in the vodka and immediately scrape up all of the browned bits from the bottom of the pot.

Continue cooking until the sharp alcohol aroma has cooked off and the mixture returns to a very thick, concentrated, scoopable texture.

Taste and adjust with kosher salt. Let the concentrate cool completely.

Divide evenly among 6 freezer-safe containers or silicone molds. Freeze until solid, then transfer the portions to a freezer bag or airtight container if desired.

Cook the pasta in generously salted water until al dente. Before draining, reserve at least 1 to 2 cups of pasta water.

Place one frozen concentrate portion in a large skillet over medium-low heat. Add a small splash of pasta water around the concentrate and let it begin melting, stirring as it softens.

Once mostly melted, stir in the heavy cream until smooth.

Add the Parmesan, then the cooked pasta.

Add reserved pasta water a little at a time, tossing continuously, until the sauce is glossy, silky and clings to the pasta. Taste and adjust with salt or additional Parmesan.

Place the frozen concentrate in a saucepan or skillet over medium-low heat. Pour about ¼ cup of the hot water around it.

As it softens, stir and scrape the melting concentrate into the water, adding the remaining hot water gradually — up to about ½ cup — until completely melted and smooth.

Stir in the heavy cream, followed by the Parmesan.

Cook gently, stirring frequently, until the sauce is smooth, glossy and cohesive. If it feels too thick, add hot water a splash at a time. Taste and season with kosher salt.

Keep it thicker and more spoonable for pizza and sandwiches; loosen it with more hot water for dipping or saucier applications.

Reheat refrigerated finished sauce gently over low to medium-low heat, adding hot water a splash at a time and stirring until it loosens and turns glossy again. If needed, add a small splash of heavy cream and a little freshly grated Parmesan. Taste for salt before serving.