Activate the yeast: Combine warm milk and yeast in a large bowl. Let sit 5 minutes until foamy.
Mix the dough: Add sugar, eggs, butter, vanilla, and salt. Mix well, then gradually add flour until a soft dough forms.
Knead: Turn dough onto a floured surface and knead for 8 minutes until smooth and slightly tacky.
First rise: Place in a greased bowl, cover, and let rise in a warm spot for 1 to 1.5 hours until doubled.
Cut: Roll dough to 1/2 inch thickness and cut into donut shapes. Place on parchment-lined baking sheets.
Second rise: Cover loosely and let rest 30 to 45 minutes until puffed.
Fry: Heat oil to 350°F. Fry donuts in small batches, 60 to 90 seconds per side until golden. Drain on a wire rack.
Make the glaze: Whisk powdered sugar, maple syrup, milk, and maple extract until smooth.
Glaze: Let donuts cool 5 minutes, then dip into maple glaze and return to the rack to set.