I'll hold the pastry-thing up, squeeze it to separate the top layer (pure pastry; ugh) from the bottom layer (carrier bed; potential), and bin the top layer, revealing the filling. In most pastries, the filling will be set along a central strip of the carrier bed layer (really, it's shocking to see how little actual filling there tends to be in these things), so I'll pick away the bare pastry, and then scrape the filling off with my front teeth, like you would if you were scraping the last of the flesh from a slice of melon. Having scraped a segment of filling off the carrier bed, I'll pick off and throw the unwanted pastry away, and then repeat the process until I've eaten all the filling, and binned all the pastry.

In fairness, there are some pastry products where I'll eat the whole carrier bed part, and a few where I'll eat the whole thing (those Jamaican pastries and fruit strudels really are the bobbins), but for low-grade pastries, I see no point eating the pastry. It just seems so gratuitously unhealthy for so little benefit.