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*Look for fresh yeast in the refrigerated or frozen food sections of gourmet markets.
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To make the brioche doughnuts: Put the warm milk in a mixing bowl (an electric mixer works best for this recipe--so if you have one put the milk in the mixer bowl).
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Sprinkle the yeast over the warm milk and allow it to dissolve.
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Whisk 1 egg and 1 cup of the flour into the yeast mixture.
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When the dough-sponge is smooth, sprinkle it with an additional cup of flour.
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Allow the dough-sponge to rise in a warm place until the top layer of flour cracks, about 30 minutes.
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Lightly beat the 4 remaining eggs.
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Then, using the dough hook attachment of an electric mixer set at medium speed, or a wooden spoon, work the eggs into the dough.
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When the dough is smooth, add the sugar, salt, and remaining 1 1/2 cups of flour all at once.
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If using a mixer, start on low and gradually increase the speed as the dough comes together, mixing for a good 15 to 20 minutes.
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If you do not have a mixer turn the dough out onto a clean, floured work surface and knead until it is smooth, elastic, and no longer sticky.
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(Don't be alarmed if the dough seems too wet.
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It will tighten up into nice, soft, elastic dough.)
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When the dough comes together, add the butter and mix for another 10 to 15 minutes.
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Cover the dough with a clean towel and set it aside in a warm place to rise until doubled in size, about 2 hours.
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Stretch the dough to release some of the trapped gasses and redistribute the yeast, cover the bowl with plastic wrap and refrigerate overnight.
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Turn the dough out onto a floured work surface.
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Gently stretch the dough into a rectangle, then roll it out about 3/4-inch thick.
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Cut the dough with a doughnut cutter.
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Transfer the doughnuts to a floured board or baking sheet.
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Cover the doughnuts with a clean towel and allow them to rise in a warm place until they feel soft and fluffy, about 1 hour.
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To make the crazy glaze: Beat the egg whites until frothy.
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Add the vanilla and confectioners' sugar and beat until smooth.
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Divide the glaze between several bowls then color with drops of food coloring.
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To fry the doughnuts: Heat 2 to 3-inches of oil in a heavy, high-sided pot over medium heat until the oil reaches 350 degrees F. Working in batches of 3 or so, drop the doughnuts into the oil and fry until they float.
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Turn the doughnuts over in the oil and continue cooking.
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Cook the doughnuts, turning them once or twice more as necessary, until they are uniformly browned, then transfer them to a plate lined with paper towels to drain.
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While the doughnuts are still warm, dip 1 side of each into the desired color of glaze then set aside on a wire rack until the glaze is firm.
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Serve warm or at room temperature.
