Into the bowl of your stand mixer, add the room-temperature milk, sugar and yeast and stir gently. Let sit for about 5 minutes.
Add the flour, salt, melted butter and egg.
Turn your mixer on low and watch the ingredients combine. After about a minute you should see a sticky dough starting to come together. If you need to turn the mixer off and scrape down the sides and bottom of the bowl, now's your chance.
Turn up the mixer speed slightly and allow the mixture to knead the dough for about 2 minutes. The dough should be tacky to the touch, but not so sticky that it's very messy. If necessary, add a tablespoon or two of extra flour. It should pull away from the sides of the bowl easily and the bowl should appear fairly clean. If the dough appears too sticky, continue kneading and add flour, one tablespoon at a time, until the dough reaches that tacky texture (don't add more than 3 tablespoons).
If you don't have a stand mixer, you can knead the dough by hand on a lightly floured work surface for 5 minutes after the wet and dry ingredients are combined.
Cover the bowl with some plastic wrap and move it to a warm place in your kitchen to rise for about 1 1/2 to 2 hours. It should double in size.
Once you notice your dough has risen fully, and before you do anything with it, grease a large rectangular pan or casserole dish (at least 9x13 in size) with some butter or cooking spray and spread the 1/3 cup of pizza sauce onto the bottom of the pan.
Turn the dough out onto a work surface that's been very lightly greased with vegetable oil. Using a rolling pin, roll out the dough evenly until you have a rectangle that's approximately 18 inches wide and 12 inches deep.
Spread the 2/3 cup of pizza sauce over the dough rectangle right to the edges. Top evenly with the chopped ham, diced bell pepper and shredded cheese.
Roll up the dough from the long side until you've got one long roll. Cut the long roll in half, then in half again, and then each of those sections into 3 pieces until you have 12 rolls. Try to cut the rolls to an even thickness as this promotes even baking. You don't want some rolls to be overdone and others to be raw in the middle.
Arrange the 12 pieces in the pan you've greased. An arrangement of 3 rows of 4 works well in a rectangular pan.
Top with the 1/2 cup shredded cheese.
Cover the pan with a clean kitchen towel and set them aside to rise. It's a good idea to preheat your oven at this time, too. Set temperature to 350 degrees Fahrenheit.
Once the rolls have risen for 20 to 25 minutes (no longer), place them in the oven to bake for about 25-30 minutes, or until they're a nice even light golden brown colour and the cheese on top is melted and bubbly.
Remove the pan from the oven and let the buns cool in the pan for about 20-25 minutes before serving. Sprinkle with some chopped chives, if desired.