In a small bowl, stir together the warm milk, warm water, and yeast. Let it sit for 5 to 10 minutes, until it becomes foamy on top.
In a large bowl, whisk the flour, sugar, salt, and cinnamon until well combined.
Add the softened butter, eggs, vanilla extract, and the foamy yeast mixture into the dry ingredients. Stir everything together until a dough begins to form.
Move the dough onto a floured surface and knead it by hand for about 8 minutes, until smooth and elastic. (You can also use a stand mixer fitted with a dough hook on medium speed for 5 to 6 minutes.)
Place the dough in a lightly greased bowl, cover it with a towel, and let it rise in a warm spot for about 1 to 1 ½ hours, or until it has doubled in size.
After the dough has risen, roll it out on a lightly floured surface until it's about ½ inch thick.
Use a donut cutter—or a large glass and a smaller one for the center—to cut out donut shapes. Lay the cut donuts onto a parchment-lined baking sheet.
Cover the tray and let the donuts rise again for 30 to 40 minutes, until they look puffy.
While the donuts are resting, pour the vegetable oil into a heavy-bottomed pot and heat it to 350°F (177°C).
Once the oil is ready, gently add 2 or 3 donuts at a time into the hot oil. Fry each side for 1 to 2 minutes until golden brown.
Use a slotted spoon to remove the donuts and set them on a wire rack lined with paper towels to drain.
In a medium bowl, whisk together the powdered sugar, maple syrup, milk (or heavy cream), vanilla extract, and a pinch of salt until smooth.
If the glaze feels too thick, stir in a little extra milk. If it’s too thin, mix in a bit more powdered sugar.
When the donuts are still slightly warm, dip the tops into the glaze.
Set them back on the wire rack and, if you like, sprinkle the tops with chopped toasted pecans and a touch of flaky sea salt.
Serve with your favorite drink.