Mussels and Fries with Mustard Mayonnaise

Cook french fries according to package instructions and keep warm in oven if necessary. Meanwhile, cut onion into very thin slices with slicer, then cook with garlic and a pinch of salt in butter in a wide heavy medium pot over medium-high heat, covered, stirring occasionally, until pale golden. While onion cooks, whisk together mustard and water until smooth, then whisk in mayonnaise and 1/4 teaspoon pepper. Add wine to onion and briskly simmer, covered, stirring occasionally, until onion is almost tender, about 5 minutes. Add mussels and cook, covered, stirring occasionally, until mussels just open wide, 4 to 6 minutes, checking frequently after 4 minutes and transferring as cooked to a bowl. (Discard any mussels that remain unopened.) Stir parsley into cooking liquid and season with salt. Pour liquid over mussels, then serve with fries and mustard mayonnaise.