Going out on a limb here but I don't think there is an English equivalent of the Italian, scarpetta, which translated in English is little shoe. One theory behind the origin of the word is that the piece of bread used to mop up the sauce acts like a shoe by scraping or sliding across the plate.
I really can't think of anything which comes close. Italians love their bread, and their pasta and will happily eat the two side by side. The British less so, in the past they would tend to butter their slices of bread or small rolls and eat them before the arrival of the main course. This was always something that struck me odd whenever I went to British restaurants. Now of course, garlic bread seems to be ubiquitous, and I'm sure Americans and British love to soak up sauces.
I'm digressing, back to to the OP question. I would use a verb to describe the action of absorbing sauce, meat juices, or indeed soup with pieces of bread.
clean or soak up (something) by wiping
He was mopping his plate with a piece of bread
Put or let something down quickly or briefly in or into (liquid)
Make or allow (something) to become thoroughly wet by immersing it in liquid