Beat eggs and sugar:

In a standing mixer bowl, beat eggs and sugar until light.

Mix wet ingredients:

Add milk, sour cream, vanilla, and melted butter; mix well.

Whisk dry ingredients:

In a separate bowl, whisk together gluten-free flour, baking soda, baking powder, salt, and xanthan gum.

Combine wet and dry:

Slowly add dry ingredients to wet ingredients, mixing until combined. Dough will be wet.

Roll dough thickly:

On a heavily floured surface (use gluten-free all-purpose or white rice flour), roll dough to 1 inch (2.5 cm) thick, using plenty of flour to prevent sticking.

Cut donuts and holes:

Cut into donuts and donut holes.

Heat oil to 375°F:

Heat oil in a deep fryer or deep pan to 375°F (190°C).

Fry donuts golden:

Fry donuts for 2-3 minutes, flipping once, until golden brown.

Drain on rack:

Transfer to a wire rack over paper towels to drain.

Make glaze:

For the glaze: whisk together powdered sugar, melted butter, and milk until smooth. Optionally, add 1/2 teaspoon vanilla or make a chocolate glaze (see note).

Glaze warm donuts:

Dip warm donuts into glaze, let excess drip off, and set on wire rack to set.