Pulse the all-purpose flour, bread flour, sugar, yeast, and salt in a food processor.

Add the cold butter and pulse 8 to 10 times until the butter pieces are about pea-sized.

Add the cold sour cream and pulse until the dough begins to clump together.

Press the dough into a disc, wrap it, and refrigerate for at least 4 hours.

Roll the dough into an 8x15-inch rectangle. Fold it like a business letter, then turn it 90 degrees, roll, and fold again.

Wrap and chill for 20 minutes. Repeat the process two more times for three total rounds.

Beat almond paste, softened butter, powdered sugar, egg white, salt, and lemon juice until mostly smooth and spreadable.

Cut the dough in half and roll one half into a 6x24-inch strip. Spread half of the filling down the center.

Fold one long side over the filling, brush with beaten egg white, and fold the other side over and pinch the seam closed.

Curve the filled strip into an oval, tuck one end into the other, and seal with egg white.

Cover the shaped pastries loosely and let rise for 30 to 45 minutes, until slightly puffy.

Preheat the oven to 375°F. Brush with egg white and bake for 25 to 28 minutes, rotating pans halfway.

As soon as the kringles come out of the oven, stack hot pans and press down gently to release trapped air.

Cool completely before icing. Whisk together powdered sugar, vanilla, lemon juice, salt, and water until smooth. Drizzle over the cooled kringles and let set.