Make the Yeast Dough
In a small bowl, dissolve the yeast and 1 teaspoon (5 ml / 0.2 fl oz) of sugar in the warm water.
Let it stand for 10 minutes until the mixture is foamy.
In a saucepan, gently heat the milk, butter, remaining sugar, and salt until the butter is just melted. Allow the mixture to cool until it's lukewarm (around 110°F / 43°C).
In a large bowl or the bowl of a stand mixer, combine the lukewarm milk mixture with about 2 cups (480 ml) of flour and beat until smooth.
Stir in the dissolved yeast mixture.
Beat in the 2 whole eggs and the egg yolk one at a time, mixing well after each addition.
Gradually add the remaining flour, mixing until a soft dough forms.
Turn the dough out onto a floured surface and knead for 10-15 minutes (or use a dough hook for 7-10 minutes) until it becomes smooth, satiny, and elastic.
Form the dough into a ball, brush it lightly with melted butter, and place it in a clean bowl.
Cover and chill in the refrigerator for at least 2 hours or overnight.
Prepare the Fillings
For the Mushroom Filling: Soak the dried mushrooms in hot water for 1 hour, then rinse them thoroughly. Wash and trim the fresh mushrooms. Finely mince all mushrooms and the bread slices using a food processor or food mill.
Melt butter in a large skillet over medium heat.
Sauté the onion and garlic until golden.
Add the mushroom-bread mixture, season generously with salt, pepper, and dill, and cook over low heat for 20-30 minutes, stirring occasionally, until the mixture is thick and the flavors have melded.
Let cool completely.
For the Cabbage Filling: In a large pan, sauté the finely chopped onion in butter until golden.
Add the shredded cabbage, salt, and pepper.
Cover and cook over low heat for about 30 minutes, stirring occasionally, until the cabbage is very tender.
Remove from heat and let the mixture cool slightly. Finely chop or press the hard-boiled eggs through a sieve and stir them into the cabbage along with the dill.
Taste and adjust seasoning.
Let cool completely.
Assemble and Bake the Pierogi
Preheat your oven to 375°F / 190°C.
Line a baking sheet with parchment paper or lightly grease and flour it.
Punch down the chilled dough and roll it out on a well-floured board to a thickness of about ¼ inch (0. 6 cm).
Use a cookie cutter or glass to cut out rounds about 3½ to 4 inches (9-10 cm) in diameter.
Place a heaping tablespoon of your cooled filling in the center of each dough round.
Fold the dough over the filling to create a half-moon shape. Pinch the edges together firmly to create a tight seal.
Place the shaped pierogi, seam-side down, on your prepared baking sheet.
Cover them with a clean towel and let them rise in a warm spot for about 30 minutes.
In a small bowl, whisk together the remaining egg and the tablespoon of milk or cream to create an egg wash.
Brush this wash gently over the tops of the pierogi.
Bake for 25-30 minutes, or until the pierogi are a deep golden brown and shiny.
Serve hot, warm, or at room temperature.
Don't overfill your pierogi. About one heaping tablespoon is perfect. Too much filling makes them difficult to seal and can cause them to burst open while baking.
