In stand mixer bowl, add 1 cup flour, yeast, sugar and salt; mix well. Add warm water and butter to flour mixture. Using a paddle attachment, blend on low speed for 30 seconds; beat on medium speed for 4 minutes.
Switch to dough hook attachment. Gradually stir in enough remaining flour to make a firm dough. Knead until smooth and elastic, about 5 to 7 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.)
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 1 hour.
Prepare filling(s): add ingredients in large bowl(s), mix to combine. Set aside.
Turn dough onto lightly floured surface; punch down to remove air bubbles. Divide dough into 6 equal parts. Divide each part into 5 pieces. Pat each piece into a 4-inch circle; place on greased cookie sheet. Spoon about 1 tablespoon filling on each circle. Fold the circle in half and seal edges by pressing with fingers, then with a fork. Brush tops with egg wash.
Preheat oven to 400°F.
Bake for 12 to 15 minutes. Remove from cookie sheet. Serve warm.