I'm trying to find the sources on youtube (last part about the English making a pie crust for throwing away)
Also would that be a sin of gluttony in wasting food?
This blog on mince pies from the organisation English Heritage contains the following passage.
Our mince pies undoubtedly have medieval origins, although we would not immediately recognise them. Pie crusts were known as coffins, and used as a vessel to cook delicate foods or house pre-boiled meat fillings. Pastry was little more than flour mixed with water to form a mouldable dough. It was designed to be discarded once the contents of the pie had been eaten, although perhaps the poor may have eaten the cast offs.
It's similar to the information in Kimchi Lovers answer, but unlike the pie merchants they have linked to English Heritage is a respected organisation responsible for protecting and researching historic sites in England such as Dover Castle, Stonehenge and Hadrians Wall.
Edit: To address the question of the seven deadly sins, it could be argued that the opposite was true. Such pies would only be made in rich households and the crusts would be passed on to the almoner for distribution to the poor hence demonstrating the Christian virtue of Charity in their minds.
It's also possible if not eaten by humans pie crusts could have been used as animal feed.
I believe you are referring to a pasty. This pie with a savory filling has been popular in England for centuries, but the method you describe of eating it and throwing away the crust is often associated with tin miners in Cornwall. The pasty was a convenient, sealed way for miners to carry a meal with them. It is alleged that the pasty was a way for them to use their hands (which were contaminated with arsenic) to eat without washing or poisoning themselves.