Warm the milk until it is comfortably warm, not hot. Combine the warm milk, instant yeast, and granulated sugar in a large mixing bowl.
Add the flour, melted butter, egg, maple syrup, and salt. Mix until a soft dough forms.
Transfer the dough to a lightly floured surface and knead until smooth and elastic, adding only a little flour if necessary.
Place the dough in a lightly greased bowl. Cover and let it rise in a warm, draft-free place until doubled in size, about 1 hour.
Roll the risen dough into a large rectangle on a lightly floured surface.
Spread the softened butter evenly over the dough. Sprinkle the brown sugar and cinnamon over the butter, then drizzle with maple syrup.
Starting from one long side, tightly roll the dough into a log. Cut into 12 evenly sized rolls using a sharp knife or unflavored dental floss.
Arrange the rolls cut-side up in a greased 9x13-inch baking dish. Cover and let them rise again until noticeably puffy, about 30–45 minutes.
Preheat the oven to 350°F (175°C). Bake the cinnamon rolls for about 22–25 minutes, or until lightly golden and cooked through.
While the rolls bake, whisk together the powdered sugar, maple syrup, melted butter, vanilla extract, salt, and milk until smooth.
Let the baked cinnamon rolls cool for a few minutes, then drizzle generously with the maple glaze.
Serve the maple cinnamon rolls warm and enjoy.