1. Make the dough: In the bowl of a stand mixer fitted with paddle attachment, combine yeast, 2 cups milk, sugar, salt, eggs, and 6 cups flour. Mix ingredients on low speed until they form a ball, about 1 minute. Increasing the speed to medium-low, slowly add butter, one pat at a time, until incorporated.
2. Using the dough hook attachment, knead on medium-low speed for 5 minutes. The dough should be soft and supple, not sticky—adjust with additional flour or milk if necessary. Continue kneading for 2 minutes more. Remove a small piece of dough. It should stretch enough that you can see through it before it tears. If you can’t, continue kneading until the dough passes this test.
3. Remove dough and place into a lightly oiled bowl and cover with plastic wrap. Let sit at room temperature until it doubles in size, about an hour.
4. Turn dough onto a lightly floured work surface and divide it into 36 pieces that are about 1½ ounces each. Moving the palm of your hand in tight circular motions, shape the dough into small balls.
5. On two greased baking trays, place balls about an inch apart from one another, brush with melted butter, and cover loosely with plastic wrap. Allow to double in size.
5. After dough balls have doubled in size, remove plastic wrap and, using a lightly floured work surface, gently flatten the ball outward into a disk. Place one piece of sausage and about a tablespoon of cheese in center of dough. Pull dough from the outside edge over the sausage to enclose the filling. Pinch the seam well and return to the baking tray. Continue with remaining dough. Brush with melted butter and preheat oven to 400 degrees.
6. Klobasniky are ready to bake when they have doubled in size and look light and puffy, about 45 minutes after filling, depending on room temperature. Brush again with melted butter and sprinkle with sea salt. Bake for 20 minutes. Rotate pan and bake for 10 minutes more. The dough should be light golden brown and between 190 and 200 degrees. Remove pan from oven and place on cooling rack. Let cool slightly before serving.