Place warm water in the bowl of a stand mixer fitted with a dough hook OR a large bowl. Dissolve yeast in water and let stand 5 minutes until foamy.
Mix honey, egg, and oil into yeast mixture until smooth. Add salt, 1 tablespoon rosemary, and 3-1/2 cups flour to bowl and mix on low speed OR with a wooden spoon if making dough by hand until dough pulls away from sides of bowl, about 2 minutes of mixing (if dough seems a bit wet, add additional flour by the tablespoon until dough pulls away from sides of bowl)
If making dough in stand mixer, knead dough in mixer on low speed 5-7 minutes until dough is stretchy and elastic. If making dough by hand, turn dough onto a lightly floured surface and knead by hand until dough is stretchy and elastic, about 10 minutes of kneading.
Place dough in a greased bowl, turning once to grease top. Cover bowl with plastic wrap and let rise in a warm place at room temperature 1 to 1-1/2 hours until dough is doubled in size.
Once dough has doubled in size, preheat oven to 350F. Lightly grease a 9x13 baking dish. Set aside.
Punch risen dough down and divide into 12 even portions. Roll each dough portion out into a smooth ball and arrange in greased pan.
Brush dinner rolls with egg white/water wash and top with asiago and remaining 1 tablespoon rosemary. Allow dinner rolls to rise in a warm place 30-45 minutes until doubled in size.
Bake dinner rolls at 350F 25-30 minutes until rolls are a deep golden brown and register 190F on a kitchen thermometer. Cool on a wire cooling rack.
In a small saucepan over medium-high heat, melt butter. Add garlic and sauté 1 minute until fragrant.
Brush garlic butter mixture over warm dinner rolls. Allow dinner rolls to cool several minutes before breaking apart and serving warm. Enjoy!