To the bowl of a stand mixer fitted with the dough hook, add 3 cups of the flour, yeast, sugar, and salt. Mix on low to combine.
Add warm milk and egg to the flour mixture. Knead for 2 minutes at medium speed, stopping to scrape the sides of the bowl as needed.
Add another ½ cup of flour and knead on medium speed for another 2 minutes.
Add the remaining ½ cup flour, garlic, chives, and shredded cheddar cheese. Knead on medium speed until the dough forms a soft, slightly tacky ball that pulls away from the sides of the bowl, about 3 to 5 minutes. (Because the cheese adds moisture, add flour 1 tablespoon at a time if the dough feels too wet, or warm water 1 teaspoon at a time if the dough feels too dry.) The cheese will form pockets throughout the dough, soavoid over-kneading, which can make the rolls tough.
Place the dough in a lightly greased bowl. Cover and let rise for 45 minutes, or until doubled in size.
Spray a 9×13-inch baking dish with nonstick cooking spray. Set aside.
When ready, punch down the dough.
Divide the dough into 12 equal-sized pieces (about 100 to 110 grams each) and shape into balls. Place into the prepared baking dish.
Cover and let rise again for 30 minutes, or until doubled and puffy.
Preheat oven to 375°F.
Whisk egg and water for egg wash; gently brush over the tops of the risen rolls.
Bake 18 to 22 minutes until golden brown.
As soon as the rolls come out of the oven, generously brush the tops with melted Cowboy Butter.
Let the rolls cool slightly, then serve warm.