Melt the butter and milk in the microwave on high for one minute or until butter is fully melted. Set this aside to cool for a few minutes.
Dissolve the sugar in the warm water in a small bowl, and add the yeast. Set this aside to proof, about five minutes, or until it is frothy and foamy.
In the bowl of a KitchenAid or similar heavy-duty mixer, blend the eggs, salt, and 1/3 cup of sugar. Whisk this together with the whisk attachment or just use a hand whisk. Attach the dough hook and add 4 1/2 cups of flour. Mix on low speed until the flour is incorporated. Add more flour in 1/4 cup increments until the dough is pulling away from the sides of the bowl (I added about 3/4 more flour during this process), and I had to scrape down the sides of the bowl a few times.
Put the dough onto a lightly floured surface and knead until smooth and elastic, adding flour as necessary. The dough should not be sticking to your hands or your counter. I added about 1/4 cup more flour during this process to get the dough where I wanted it.
Lightly oil a bowl that's at least twice the size of the ball of dough. Put the dough in the bowl and turn it around until sheened with oil. Cover and let rest in a warm spot for an hour or until it doubles in size.
On a clean, lightly floured surface, divide the risen dough in quarters. Roll each quarter of the dough into a long log and cut it into six pieces. Roll each piece into a ball. Each ball should be slightly larger than a golf ball.
Roll each ball of dough into a circle that's a bit longer than the piece of sausage. don't roll the dough too thin - you don't want holes in your dough. Working one at a time, roll the dough, then lay the sausage on the dough, add a half a slice of cheese (optional) and roll the kolache up burrito-style, tucking in the ends as you go.
Lay the kolache on a parchment-lined baking sheet and when they're all done, let them rise for another 45 minutes.
Melt a few tablespoons of butter and brush the tops of your risen kolaches with butter. Bake them at 375-F for about 12 minutes, or until they're golden brown and not gooey inside.
Enjoy warm from the oven, or freeze some to heat up for mornings on the go!