Do not throw this away: it is precious liquid (see recipe tip). Add the butter to the hot, emptied-out potato pan, and let it start melting, then tip in the drained potatoes and garlic.
add the garlic and bring to a very gentle simmer, stirring constantly. Add around 100ml/3½fl oz of the pasta cooking water and the grated Parmesan and stir until melted. Season with salt and pepper.