Make the dough by mixing all the ingredients together in a large bowl. Knead the dough until it comes together as a sift, silky dough, about 5 minutes. Cover in plastic wrap and let the dough sit on the counter for an hour or so. You can refrigerate it up to 2 days.
To make the filling, chop the fresh and dried mushrooms and add them to a large saute pan with the minced onions. Turn the heat to medium-high and saute until the mushrooms give up their water, about 2 to 5 minutes depending on the mushroom. Salt them well. When the water from the mushrooms has almost evaporated, add the butter and saute until everything begins to brown, about 4 to 6 minutes. Add the garlic and cook another minute. Turn off the heat and let the mixture cool for a few minutes.
Add the contents of the pan to a food processor, along with the dill or parsley. Buzz to make a fine crumble, but not a paste. You want some texture in the filling. Put the contents of the food processor in a bowl and mix well with breadcrumbs. Add salt and black pepper to taste.
To make the pierogi, roll the dough out by hand -- it's too soft for a pasta roller.
And you don't want it too thin, either. About 1/8 inch is good. The reason is you stretch the dough when you make the pierogi.
I use pastry cutters to make my pierogi, but you can use a glass or you can cut out squares. How wide is up to you. I used a 3-inch round, which makes smallish pierogi that are a bit more like potstickers in size. Everyone's pierogi are different shapes and sizes, so go for whatever floats your boat.
Take the round of dough and stretch it a little evenly all around, like you would with pizza dough. It doesn't have to be too stretched out, but a little helps. Hold the dough in the palm of your "off" hand. Put a teaspoon, tablespoon or whatever seems to fit in the center of the dough -- you can fill pierogi more than you can with Italian pasta dough because it's so flexible.
Fold the dough over the filling. I start at the center and work towards an edge, stretching and pinching the dough at the rim of the dumpling as I go. Finish by sealing the other edge. Set the dumpling on your well-floured surface and seal it tight by crimping with the tines of a fork. Set your finished pierogi on a baking sheet that you've dusted with either semolina flour or cornmeal.
Pierogi are pretty moist, so they don't do too well stored uncooked, unless you freeze them right away. To freeze, put the baking sheet in the freezer (or a plate if the whole sheet won't fit), then, when they are frozen, you can put them in a freezer bag. Normally I cook pierogi shortly after making them.
You can either simply boil your pierogi until they float -- I give them another minute once they float, too -- or you can double cook them, which is what I prefer. To do this, boil until they float, then fry in lots of butter. Serve with caramelized onions, sour cream and dill.