Whisk together Buttermilk (3/4 cup) and Water (1/4 cup). Warm to 95-110 degrees F (35-40 degrees C) in the microwave using 30-second intervals.
Combine warmed liquids, Granulated Sugar (1 teaspoon), and Active Dry Yeast (1 tablespoon) in a stand mixer bowl fitted with a dough hook. Allow yeast to bloom for 10 minutes.
Stir in 1 1/2 cups of the All-Purpose Flour (2 1/4 cup), the Fresh Rosemary (1 tablespoon), and the Kosher Salt (1 teaspoon).
Start kneading and add the Unsalted Butter (1/4 cup) a few chunks at a time, along with additional flour, until the butter has been incorporated and the dough is starting to pull away from the sides. Continue to add flour by the tablespoon until the dough is smooth and no longer sticks to the side of the bowl, about 8 minutes.
Place dough in a lightly oiled bowl and cover with plastic wrap or kitchen towel. Allow dough to rise in a warm, draft-free area until doubled in size, approximately 60 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Remove dough from bowl to a clean work surface. Knead 2-3 times by hand to remove the air.
Divide dough into 12 equal portions of 55-58 grams each. Roll into balls and place on a parchment lined baking tray 1/4-inch apart. Cover with oiled plastic wrap or kitchen towel and allow to rise for 45 minutes.
Whisk Large Egg (1) and Water (1 tablespoon) with a fork and brush evenly onto rolls. Bake in preheated oven for 18-20 minutes, rotating tray halfway through baking time.
Remove from oven and slide onto a cooling rack, removing the parchment paper. Allow rolls to cool to slightly warm or room temperature for serving.