Generously butter a large bowl. Set aside. Place 1 cup milk in a small saucepan, over medium heat. Heat until a candy thermometer registers 110 degrees F. Pour milk into bowl of an electric mixer fitted with the dough hook. With machine on low speed, add granulated sugar, yeast, 2 teaspoons salt, butter, cinnamon, nutmeg, and whole eggs. Add flour, 1 cup at a time, and mix on low speed until a soft, slightly sticky dough forms around dough hook. Continue kneading, scraping down hook and sides of bowl as necessary, until smooth, about 4 minutes more.
Add currants; knead, with dough hook, to incorporate. Turn dough out onto a heavily floured surface. Knead briefly to evenly distribute currants in dough. Shape into a ball. Place dough in prepared bowl. Turn to coat with butter. Cover with plastic wrap. Let rise in a warm place until doubled in bulk, about 1 hour and 20 minutes.
Generously butter an 11 by-17 inch baking pan. Set aside. Turn dough out onto work surface. Knead briefly and divide dough into 24 equal pieces (about 2 ounces each). Shape each into a tightly formed ball. Place on prepared pan, spacing 1/2-inch apart. Cover with plastic wrap. Let rise in a warm place until buns are touching and doubled in bulk, about 1 hour.
Heat oven to 375 degrees F with rack positioned in center.
In a small bowl, whisk together 1 egg white, 1 tablespoon water, and a pinch of salt. Brush tops of buns with egg wash. Using scissors or a very sharp, buttered slicing knife, cut a cross into the top of each bun.
Bake until golden brown, 18 to 25 minutes. Rotate front to back after 10 minutes to ensure even color. Transfer pan to a wire rack. Cool to room temperature, about 30 minutes. Whisk together remaining tablespoon milk, confectioners' sugar, and lemon juice. Pipe or ladle glaze over buns.
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