In the bowl of a stand mixer, fitted with the dough hook attachment, add the flours.
Whisk in the sugar and the salt, and then whisk in the yeast.
Turn the mixer on low and add the water, milk and egg.
Increase speed to medium and add softened butter.
Beat until smooth and elastic, about 5 minutes. (You can do this by hand, too, but if take about 20 minutes of kneading).
Place dough in an oiled bowl and cover with plastic wrap and let rise until double in size, usually an hour to 2 hours, but could take longer. (You can also place in the refrigerator for overnight).
On a floured work surface, roll out the dough to 1/2-inch thickness.
Using a 3-inch cookie cutter, or an inverted metal can, or an inverted glass, cut rounds from the dough.
Re-roll the scraps, and cut more rounds.
Place each round on a parchment paper-lined baking sheet, about 1 inch apart from each other, cover with a damp towel and let rise at room temperature until nearly doubled in size, about 45 minutes to an hour.
Pre-heat oven to 375°F.
Using your thumb, press 1 large indention into the center of each round, leaving a 1/2-inch wide rim.
Spoon about 1 tablespoon topping into the indentions.
Bake until golden brown, about 15 to 25 minutes (depending on the size of the kolaches).
Transfer to a wire rack and let cool