Combine the warm milk, warm water, and yeast in a small bowl. Let sit for 5–10 minutes, until foamy.

In a large bowl, whisk together the flour, sugar, salt, and cinnamon. Add the softened butter, eggs, vanilla, and yeast mixture. Mix until a dough forms.

Knead the dough on a lightly floured surface for about 8 minutes, until smooth and elastic. Alternatively, knead with a stand mixer and dough hook for 5–6 minutes on medium speed.

Transfer the dough to a lightly greased bowl, cover, and let rise in a warm place for 1–1½ hours, or until doubled in size.

Roll the dough out on a lightly floured surface to about ½-inch thick. Cut into donut shapes and place on a parchment-lined baking sheet.

Cover the donuts and let rise for another 30–40 minutes, until puffy.

Heat the vegetable oil in a heavy-bottomed pot to 350°F (177°C). Fry 2–3 donuts at a time for 1–2 minutes per side, until golden brown. Transfer to a wire rack lined with paper towels to drain.

Whisk together the powdered sugar, maple syrup, milk or heavy cream, vanilla, and a pinch of salt until smooth. Add more milk if the glaze is too thick or more powdered sugar if it is too thin.

Dip the tops of the slightly warm donuts into the maple glaze, then place them on a wire rack.

Sprinkle with toasted pecans and let the glaze set slightly before serving.