Add the warm milk, yeast, and sugar to a large bowl. Set aside until the yeast has bloomed/bubbled. About 10 minutes.
In the bowl of a stand mixer fitted with a dough hook, add the flour and salt and mix.
Transfer the bloomed yeast, 1 egg, and 3 tablespoons of butter to the stand mixer. Mix on low speed for 10 minutes. The dough should be smooth and elastic (it should bounce back when you poke it).
Transfer the dough to a lightly oiled bowl and cover with plastic wrap and a tea towel. Place in a warm place to rise until doubled in size. About 1.5 hours.
Cut the cream cheese into 12 same-sized pieces.
Assemble the rolls. Once the dough has doubled in size, turn it out onto a lightly floured surface. Cut it into 12 pieces, roughly 2-2.5 oz. each.
Place a piece of dough onto the work surface, cup your hand, and roll until a rough ball has formed. It will have some creases and won’t be perfect. Thats ok. Continue until all dough balls have formed.
Take one dough ball on the work surface and gently press it into a circle. Place a piece of cream cheese into the center and fold the edges of the dough over to cover. Pinch the edges together to seal the dough close.
Spread the remaining tablespoon of butter into a 9x13-inch casserole dish.
Place the dough balls seam-side down in the casserole dish. Whisk the remaining egg and brush the tops with the whisked egg. Allow to rise for 30-45 minutes.
Preheat the oven to 350℉. Bake for 18-20 minutes.
While the rolls are baking, place all of the topping ingredients into a saucepan. Heat until the butter has melted. About 5 minutes.
When the rolls are done, brush them with the garlic chive melted butter.