Place the water and parmigiano-reggiano rinds in a saucepan with a small pinch of salt. Bring to the barest of simmers and let the cheese rinds infuse in the water for at least 30 minutes, more if you want a more assertive cheese taste.

In a non-stick skillet large enough to hold all the pasta nests snugly in a single layer, place a knob of butter and a ladleful or two of the cheese broth.

Bring the broth to a lively simmer and add the tagliatelle. Let the tagliatelle simmer in the broth, loosening the nests with tongs or a wooden spoon as the pasta softens.

When the broth evaporates, add another ladleful to the skillet. Repeat as needed, stirring the pasta from time to time.

When the pasta is done to your liking and most but not all of the broth has evaporated, lower the heat as low as it will go and add the rest of the butter in chunks. Stir vigorously to melt the butter and emulsify it into the broth. Taste and adjust for seasoning.

Now turn off the heat and add the grated cheese. Give the pasta another vigorous stir, until the cheese has completely melted to form a creamy sauce.

Serve immediately. Have some more grated cheese on hand for those who want it.