Warm the milk in a small saucepan over medium heat just until it starts to bubble. Remove from heat right away. Stir in the sugar, ¼ cup of butter, and salt. Mix until the sugar dissolves, then set aside to cool to room temperature—about 10 minutes.

In the bowl of a stand mixer, combine the warm water and yeast. Stir until dissolved. Add 2 cups of flour, the cooled milk mixture, and the beaten eggs. Using the dough hook, mix until smooth. Gradually add the remaining flour while mixing until the dough becomes elastic and slightly stiff, but not dry.

Transfer the dough to a floured surface and knead for 10 to 15 minutes, or until very smooth and stretchy. Lightly brush the dough with some of the melted butter, place it in a bowl, and cover with plastic wrap or a damp towel. Let it rise in a warm spot until doubled in size, about 1 hour.

While the dough rises, slice each sausage link in half and pat them dry—this step matters, so don’t skip it. Then, slice the block of Cheddar cheese from the short end, making pieces roughly the same length as the sausage halves.

Preheat the oven to 350°F (175°C) and lightly grease a baking sheet.

Once the dough has risen, turn it out onto a lightly floured surface. Roll it into a log and divide it into 5 equal sections. Cut each section into 4 pieces, making 20 total. Flatten each piece into a circle or oval with your palm. Place a slice of cheese and a sausage half in the center, then wrap the dough around the filling. Pinch the seams shut tightly and smooth them out. Place each kolache seam-side down on the baking sheet. Repeat with the remaining dough and filling.

Brush the tops with the remaining melted butter and bake for 12 to 15 minutes, or until golden brown.