Measure out the milk and water in a microwave safe measuring cup or saucepan. Heat up the mixture, either in the microwave or in a saucepan, until it is "fingervarm" (finger warm), meaning it should feel warm to the touch but not hot. You don’t want to kill the yeast! Add the yeast and sugar to the liquid and stir to dissolve. Let it rest 5-10 minutes until it begins to froth and bubble a little bit. (If your yeast doesn’t do anything, it may be expired. Check the expiration date on the package and start over with new yeast!)
Add the salt and flour to a large mixing bowl or the bowl of a stand mixer. Add the melted butter, and the water/milk/sugar/yeast mixture and knead for 8-10 minutes (either using the dough hook attachment on the mixer or kneading by hand) until smooth and elastic. Note: you may have to add a little more water or a little more flour if your dough is either too dry or too sticky. Adjust until the dough releases from the edges of the bowl and there are no dry bits at the bottom of the bowl!
Prepare your fillings by grating the cheese and chopping the ham into cubes (if needed) and mincing up the chives.
Add the fillings to the stand mixer or the bowl and knead until they are evenly distributed, either using the mixer or by hand.
Leave to rise in a warm place for ~45 minutes to an hour or until doubled in size.
Once the dough has doubled in size, remove from the bowl onto a clean work surface. Divide the dough into 12 equal pieces. Shape the rolls by taking each piece, then cup it in your hand with the top of the roll touching your palm. Place your hand on an unfloured surface with the seam touching the table. Move your cupped hand in quick, tight circles – this will cause the seam to come together and create tension on the surface of the dough. Repeat with all the pieces, and place on a baking sheet lined with parchment paper or a silicone mat.
Allow to rise for another 30-45 minutes in a warm place until doubled in size. In the meantime, preheat the oven to 375 F (190 C).
Once the rolls have risen, brush the tops with milk. Bake for ~15 minutes or until lightly golden with brown specks. Let them cool slightly, but they are best enjoyed warm! We store these in the fridge and reheat before eating.