Whisk together the yeast, water and 1/2 tsp of the granulated sugar and let rest for 5 minutes in an electric stand mixer bowl.
Set mixer with paddle attachment and mix in remaining sugar (2 Tbsp + 2 1/2 tsp) milk, egg, butter, rosemary, garlic and salt on low speed (the butter won't blend until the flour is mixed it). Add 2 cups of flour and mix on low speed until combined, then switch to a hook attachment, set mixer on low speed and slowly add in remaining 1 1/2 cups flour. Allow mixture to knead on medium-low speed until smooth and elastic, adding more flour as needed (you shouldn't need more than an additional 1/4 - 1/2 cup, dough should be slightly sticky. If too much flour is added rolls will be dense and heavy).
Cover bowl with plastic wrap and rest 15 minutes. Meanwhile butter a 13 by 9-inch baking dish. Punch dough down and divide into 12 equal portions, shape each into a ball (I like to pull the sides down and tuck under several times then make into a ball shape). Cover with plastic wrap and allow to rise in a warm place about 45 - 60 minutes until nearly doubled. Preheat oven to 350 during last 10 minutes of dough rising.
Gently brush tops of rolls evenly with 1 Tbsp whisked egg. Bake in preheated oven until tops are golden brown, about 23 - 26 minutes. Serve warm, reheat before serving as necessary.