To me, 'dough pockets' is almost as bad as 'paste bags.' Dough is the raw, uncooked stuff you roll out to make pasta (or bread or cookies or pie crust or pizza), not the pasta after it has been formed into shapes, much less after it has been cooked (and ready-made filled pasta in grocery stores is often pre-cooked).
'Pastry' is different, since that's not noodle-like (wet, boiled, flexible) but bread- or crust-like in texture (dry, oven-baked, crisp, flaky). And 'pasty' is only British for a particular meat-filled pastry AFAIK.
In English we would probably just use the name of the pasta: tortellini, calzone, ravioli, etc. If it was necessary to give an explanation in parentheses for an unfamiliar kind of pasta, anything along the lines of
pasta with (meat, cheese, etc.) stuffing/filling
would probably work, but I'm not sure any of those really justifies a dictionary entry.
What's the original word again? If it's Russian, it might well be used in English too, at least in some contexts such as delicatessens, or some parts of the country with a lot of Russian residents.