To the bowl of a stand mixer fitted with the dough hook, combine 3 cups of flour, yeast, sugar, egg, and kosher salt.
Warm the milk to 110°F–115°F and add it to the flour mixture. Knead on medium speed for 2 minutes, stopping to scrape the bowl as needed.
Add ½ cup of flour and knead for another 2 minutes on medium speed.
Add the remaining ½ cup of flour along with the cheddar cheese, garlic, and chives. Knead until the dough is soft and pulls away from the sides of the bowl. If the dough is too sticky, add extra flour ¼ cup at a time, kneading after each addition until the dough pulls cleanly away from the sides.
Cover the bowl with plastic wrap and let the dough rise for 45 minutes, or until doubled in size.
Punch down the dough and prepare a 9×13-inch baking dish by spraying it with nonstick cooking spray.
Divide the dough into 12 equal pieces and shape them into balls. Arrange the balls in the prepared baking dish, leaving space between them. Cover with plastic wrap and let them rise for 30 minutes, or until doubled in size.
Preheat the oven to 375°F.
Bake the rolls for 22–27 minutes, or until golden brown.
In a small bowl, stir together the melted butter, garlic, and chives. Brush the warm rolls with the garlic butter.
Sprinkle the tops of the rolls with kosher salt and serve warm.