In a large bowl, combine the flour, sugar, baking powder, salt, cinnamon, and nutmeg. Mix well.
In another bowl, whisk together the milk, egg, melted butter, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix; small lumps are okay.
Heat oil in a deep fryer or heavy pot to 350°F (175°C).
Using a small cookie scoop or two teaspoons, drop balls of dough into the hot oil, being careful not to overcrowd the pan. Fry them for about 3-4 minutes or until golden brown, turning occasionally for even cooking.
Remove the donut holes from the oil using a slotted spoon and place them on a paper towel-lined plate to drain excess oil.
In a separate bowl, whisk together powdered sugar, maple syrup, and enough milk to achieve a smooth glaze consistency.
Once the donut holes are cool to the touch, dip them into the maple glaze, allowing any excess glaze to drip off. Place them on a wire rack to set.
Serve warm or at room temperature, and enjoy the sweet, rich flavor of maple glaze with each delightful bite!