Make the tangzhong: Whisk milk, flour, and water in a small saucepan until smooth. Cook over medium heat, whisking constantly, until a thick paste forms, about 1-2 minutes. Cool until just warm to the touch.
Make the dough: In a stand mixer bowl, whisk flour, yeast, salt, sugar, and cardamom together by hand. Add the warm milk, egg, tangzhong and melted butter. Knead on medium low speed 5–6 min until the dough is smooth, elastic and slightly tacky. Cover and let rise at room temperature until doubled in size, 1 ½ to 2 hours.
Shape the loaf: Divide the dough into 3 equal pieces. Roll each piece into a long rope, 16-18 inches long, and gently press the ropes together at the top. Braid the ropes together loosely, pinching and tucking the ends under the loaf to finish. Transfer the braided loaf to a parchment-lined baking sheet. Cover and let rise until puffy, 45-60 min. Meanwhile, preheat the oven to 350°F.
Make the egg wash and bake: Whisk egg with water, brush over the loaf and sprinkle with pearl sugar. Bake until set and golden brown, about 30 minutes. Brush with melted butter and cover with a tea towel (a non-terrycloth kitchen towel) for 20 minutes. Uncover and cool completely before slicing and serving.
Notes
To use active dry yeast in place of the instant yeast - Increase the amount to 2 ½ teaspoons and proof in the warm milk for about 10 minutes prior to adding to the recipe.If using pre ground-cardamom - Reduce the amount of cardamom to 2 - 2 ½ teaspoons.Glazed pulla variation - Brush the bread with the egg wash but skip the pearl sugar prior to baking. After the bread has cooled, combine 1 cup powdered sugar and 1-2 tablespoons of milk or water. Whisk until smooth, adding more milk/water as necessary to create a pourable glaze. Add a splash of vanilla or almond extract if desired. Drizzle in a zig zag motion over the bread. Let the glaze set slightly prior to serving.To store - Once the bread has cooled completely, wrap tightly in plastic wrap and store at room temperature for 2-3 days. Freeze for longer storage.To make ahead - Once you braid the loaf, transfer it to a baking sheet and cover tightly with plastic wrap. Store in the refrigerator overnight. In the morning, take the loaf out of the refrigerator and remove the plastic wrap. Loosely cover the loaf with plastic wrap or a tea towel and let rise at room temperature for 1 ½ to 2 hours. Brush with egg wash, sprinkle with pearl sugar and bake as directed.