For dough, in a large mixing bowl, stir together flour and salt. Cut cold butter into 1/2-inch-thick slices (not cubes). Add butter slices to flour mixture; toss until slices are coated and separated.
In a small mixing bowl, stir together egg and ice water. Add all at once to flour mixture. Using a spoon, quickly mix (butter will remain in large pieces and flour will not be completely moistened).
Turn the dough out onto a lightly floured pastry cloth. Knead the dough 10 times by pressing and pushing dough together to form a rough-looking ball, lifting pastry cloth if necessary to press the dough together. Shape the dough into a rectangle (dough still will have some dry-looking areas). Make corners as square as possible. Slightly flatten dough. Working on a well-floured pastry cloth, roll dough into a 15x10-inch rectangle. Fold 2 short sides to meet in center; bring top edge down to meet bottom edge to form 4 layers each measuring (rectangle will now measure 7-1/2x5 inches).
Repeat the rolling and folding process once more. Wrap dough with plastic wrap and chill for 20 minutes. Repeat rolling and folding process 2 more times. Chill dough for 20 minutes before using.
For filling, in a small bowl, stir together egg white, almond paste, 1/2 cup granulated sugar and the brown sugar. Set aside.
Using a sharp knife, cut dough crosswise into 4 equal portions. Wrap 3 portions in plastic wrap and chill. On a well-floured surface, roll 1 portion into a 12-1/2x10-inch rectangle. Cut rectangle crosswise into five 10x2-1/2-inch strips.
Shape a slightly rounded tablespoon of the filling into a 9-inch-long rope and place it down the center of one strip. Roll up the strip lengthwise. Brush edge and ends with water; pinch to seal. Place, seam side down, on an ungreased baking sheet, shaping strip into a letter (traditionally the letter "S"). Brush with water and sprinkle with additional granulated sugar. Repeat with remaining dough strips and filling. Repeat with remaining dough portions and filling.
Bake in a 375 degree F oven for 20 to 25 minutes or until golden. Remove from baking sheet. Cool on wire racks.
For best results, use an almond paste made without syrup or liquid glucose.