In a medium bowl, dissolve the yeast in the warm water. When dissolved, mix in 1/2 cup of the flour with your fingers, rubbing them together to break up any lumps. Cover with a barely dampened towel and let rise in a warm place until doubled in bulk, about 30 minutes. Meanwhile, in a large bowl, combine the remaining 1 3/4 cups of flour and the salt. Add the yeast mixture and mix together with the tips of your fingers. Gradually sprinkle in the lukewarm water, mixing with your fingers to make a pliable dough. If the water is all mixed in and the dough still seems dry, add another tablespoon of water. Knead until smooth, about 4 minutes. Return the dough to the large bowl, cover with the damp towel, and let rise in a warm place until doubled in bulk, about 1 hour. Turn the dough out onto a floured work surface and dust the top with flour. Roll into a rectangle about the same size as a sheet of paper (8 1/2 by 11-inches). Turn the rectangle so that it is vertical to your body. Dot the lower 2/3 of the dough with the pieces of butter and sprinkle with sugar. Working as though you are folding a letter, and folding the top down first, fold the dough in thirds. Roll out the folded dough into a rectangle, 8 1/2 by 11 inches. Fold in 3 again in the same way, cover with the damp towel, and let rest 15 minutes. Roll into a rectangle again, fold in 3, cover with the damp towel, and let rest 15 minutes. One last time, roll into a rectangle, fold in 3, cover with the damp towel, and let rest 15 minutes. Preheat the oven to 400-degrees F. Butter and flour a 9-inch cake pan. Roll the dough into a circle about 9 inches in diameter. Transfer to the prepared pan. Using a sharp knife or razor blade, make a cross-hatch pattern across the top of the cake (like a tic-tac-toe board). Bake for 20 minutes, brushing the top every 5 minutes with the melted butter that oozes out from the cake. Sprinkle with the confectioners' sugar and continue baking (do not baste) until golden brown on the top but still moist inside, about 10 minutes more. Let cool on a rack until the crust is caramelized and slightly crispy. Wash strawberries in cold water, allow the berries to drain. Combine the egg white and water and foam up mixture with a whisk. Dip the strawberries in the foam and then dip in superfine sugar. Arrange the frosted strawberries in the center of the cake. Serve at room temperature or slightly warm, cut into wedges.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.