I start by combining the softened butter, sugar, eggs, ground almonds, flour, vanilla extract, and almond extract in a mixing bowl. I beat everything together for about two to three minutes until the mixture turns light, fluffy, and well combined. Once it looks smooth and pale, I transfer it into a ziplock bag and set it aside until I'm ready to fill the dough.
I let my puff pastry sheets thaw on the counter just until I can unfold them without cracking. For one large braid, I stack both sheets together and roll them out to roughly 16 by 12 inches. If I want two smaller braids instead, I roll each sheet separately to about 12 by 12 inches. Keeping the dough cold while working helps it puff up beautifully in the oven.
Next, I score the pastry along both long edges into one-inch strips, being careful not to cut all the way through the center section. I mentally divide the dough into thirds lengthwise, then snip the corner of my filled ziplock bag and pipe the almond cream down the middle third in an even line.
Using a sharp knife, I cut through the strips along the scored lines on the unfilled sides. Then I fold the strips over the filling one at a time, alternating from the right side and the left side, working my way down until I reach the end. I press the top and bottom edges firmly so the filling stays sealed inside while baking.
Before baking, I like to freeze the shaped braid for about an hour so the filling doesn't leak out in the oven. I whisk together the egg and water, brush this wash over the chilled braid, and sprinkle it generously with granulated sugar. It bakes in a preheated 400°F oven for 30 to 35 minutes, until it turns deep golden brown and puffs up nicely in the center. If it browns too fast, I loosely tent it with foil.
While the braid bakes, I whisk together the powdered sugar and half the milk, adding more milk gradually until the glaze is thick but still pourable. Once the braid is fully baked, I let it cool slightly on a rack, trim away any filling that oozed out for a cleaner look, and drizzle the glaze over the top in a zig-zag pattern before slicing and serving.