To the bowl of a stand mixer with the dough hook attachment, add 3 cups of flour, yeast, sugar, and salt. Mix to combine.
Add warm milk and egg to the flour mixture. Knead for 2 minutes at medium speed.
Add another ½ cup of flour. Knead at medium speed for another 2 minutes.
Add the remaining ½ cup of flour, garlic, and chives. Knead until the dough pulls away from the sides of the bowl. If the dough seems too wet, add more flour in ¼ cup increments until the dough has pulled away from the sides of the bowl while kneading. If the dough is too dry, add more water in ¼ cup increments.
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 and shape into balls. Place into the prepared baking dish.
Cover the rolls and let them rise for 30 minutes, or until doubled in size.
Preheat the oven to 375°F.
In a small bowl, whisk together egg and water to make an egg wash. Gently brush the egg wash over the tops of the risen rolls.
Bake for 18 to 22 minutes, or until the rolls are 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.