In a cup, combine the yeast and 1/2 teaspoon granulated sugar. Add the warm water and stir let stand until foamy, about 5 minutes. Combine 1 1/2 cups flour, 4 tablespoons granulated sugar and the salt in a large bowl.
Melt the butter. Using an electric mixer set on low speed (whisking works fine, too) beat in 1/2 cup milk, 3 tablespoons butter, the egg and the yeast mixture. Beat on medium speed for 2 minutes. Stir in 1 1/2 cups flour to make a soft dough.
Knead the dough until smooth and elastic, about 10 minutes, adding remaining flour if needed. Cover; let sit for 10 minutes.
In a small bowl, combine remaining granulated sugar with the almonds, brown sugar and cinnamon.
Roll the dough into a 12- x 10-inch rectangle. Brush with most of the remaining butter and sprinkle the sugar mixture on top.
Roll the dough up tightly from one side. Cut the rolled dough into 12 slices. Place in a 13- x 9-inch nonstick baking pan. Brush with the rest of the butter. Cover; let buns rise in a warm place for 30 minutes. Preheat oven to 350°F Bake for 20 minutes.
Cool briefly. Mix confectioners' sugar, vanilla and remaining milk to create the glaze. Drizzle over the warm buns.