For the doughnuts: Sprinkle the yeast over the warm water in a medium bowl. Stir a few times and let stand to allow the yeast to dissolve and begin bubbling, 3 to 5 minutes. Sift 1 cup of the flour into the bowl and stir until mostly smooth. Cover with plastic wrap and set aside in a warm place for about 30 minutes. The mixture will expand, loosen in texture and will have large bubbles on the surface.
Beat the granulated sugar and butter with an electric mixer on medium-high speed until light and fluffy, 2 minutes. Beat in the vanilla, salt, nutmeg and eggs until smooth. Beat in the yeast mixture and then reduce the speed to medium-low and gradually beat in another 2 cups of the flour. Gradually add the remaining 3/4 cup of flour, 1/4 cup at a time. Scrape down the sides and bottom of the bowl as you go, until the dough is no longer sticky. It will not come together in a clean ball and you may not need all the flour.
Transfer the dough to a lightly greased large bowl and cover with plastic wrap. Let rise in a warm, draft-free place until the dough has doubled in size, about 1 1/2 hours.
Generously dust a large cutting board or counter with flour and scrape the dough out of the bowl. Dust the top of the dough with flour and pat to about 3/8-inch thick.
Cut rounds from the dough using a 2 3/4- to 3-inch biscuit or round cookie cutter. Place the cut outs on a floured baking sheet, leaving about 3 inches between each doughnut, and lightly cover with plastic wrap. Gently knead the dough scraps together, reroll and cut out the remaining doughnuts.
Let the cut doughnuts rise until well puffed but not quite doubled, another 1 1/2 hours. When the doughnuts are close to being ready, bring 2 inches of vegetable oil to 350 degrees F in a heavy-bottomed pot or Dutch oven.
Using your fingers, firmly press a well into the center of each doughnut about 1-inch wide. Fry the doughnuts, a few at a time, turning occasionally, until browned, puffed and cooked through, 3 to 4 minutes. Drain on a wire rack set over a rimmed baking sheet and cool, 10 minutes.
While the doughnuts cool, make the toppings.
For the chocolate glaze: In a medium bowl combine the confectioners' sugar, cocoa, corn syrup and vanilla. Add the milk, 1 tablespoon at a time, whisking thoroughly, until the glaze is thick, shiny and smooth. If the glaze gets thick or hard on top as it sits, whisk it back together adding a few drops of milk to smooth it out again.
For the peanut butter frosting: Beat the peanut butter and butter until smooth. Add the confectioners' sugar and vanilla and beat until thick and fluffy. Gradually beat in the milk until the frosting is spreadable but still thick.
To assemble the doughnuts, dip the tops of the doughnuts (the side with the indent) in the chocolate glaze, allow some of the excess glaze to drip off and set the doughnut back on the wire rack for a few minutes, glaze-side up. Once the glaze has set, spoon a few teaspoons of the peanut butter frosting in the center of the doughnut, spreading it and filling the well, to create a buckeye.