Boil your pasta in well salted water ntil al dente. (For extra flavor, use the same water you boiled the asparagus in, topped up if need be.)
For freshly made egg pasta, just let it come back to the boil. If your pasta has been rolled really thin, that should be enough. Otherwise, you can let it go for a minute or so, but no more. If using store-bought tagliatelle, follow the instructions on the box.
When the pasta is cooked, drain and transfer it to the skillet, along with a small ladleful of the pasta water.
Mix the pasta with the asparagus purée and tips over low heat. Let the pasta absorb the purée until it is well coated but slight quite slithery. You can add more water if things feel sticky.