Remove the silky filaments from the mussels and, holding them under running water, scrub them with a hard brush or scrape them with a knife. Rinse several times, put them without seasoning or water in a pot with a lid and cook over high heat until they all open.
Drain, conserving the cooking liquid, shuck half of them and remove the mussels from the other half so that each half-shell holds two mussels. Put the grated breadcrumbs in a bowl with the grated garlic, a handful of minced parsley, the olive oil and a small portion of the cooking liquid and mix to make a dense paste.
Cover the two mussels in each half-shell with the paste, pressing down to compact the filling, and as they are ready line them up on a baking sheet. Pour a drop of oil on each one and bake in the oven preheated to 200°C (400°F) for about ten minutes, until golden. Let them cool somewhat prior to serving.