In a large bowl, whisk together the gluten free flour, sugar, yeast, salt, and nutmeg. In a separate bowl, combine the warm milk, eggs, melted butter, and vanilla. Pour the wet mixture into the dry ingredients and stir until a soft, sticky dough forms. Cover the bowl with plastic wrap and let the dough rise in a warm spot for about 1 hour, or until doubled in size.
Turn the dough out onto a well-floured surface using gluten free flour. Gently pat it out to about 1/2-inch thickness. Use a donut cutter or two round cutters (one large, one small) to cut out donut shapes. Gather the scraps, pat out again, and cut more donuts. Place the donuts on a parchment-lined baking sheet.
Cover the cut donuts loosely with a clean kitchen towel and let them rise for another 30 minutes, until slightly puffy and pillowy.
Heat about 3 inches of vegetable oil in a large heavy-bottomed pot to 350 degrees F. Carefully lower 2 to 3 donuts at a time into the oil. Fry for about 1 to 2 minutes per side, until deep golden brown. Transfer to a wire rack set over a baking sheet and let them cool for a few minutes.
While the donuts cool slightly, whisk together the powdered sugar, milk, and vanilla until smooth and pourable. If the glaze is too thick, add milk a teaspoon at a time.
Dip the top of each warm donut into the glaze, letting the excess drip off. Set them back on the wire rack and let the glaze set for about 5 minutes before serving.