Heat the milk and butter in a medium saucepan over medium heat until the butter is melted and the mixture reaches approximately 125°F. Remove from heat.

In a stand mixer fitted with a paddle attachment, combine 2-1/4 cups of flour, sugar, salt, and yeast. Slowly add the hot milk mixture and beat on medium speed until well combined. Add the eggs and vanilla, beating for 2-3 minutes until fluffy. Add the remaining 1 cup of flour and beat until smooth.

Scrape the dough into a lightly greased bowl and cover with a clean kitchen towel. Allow the dough to stand in a warm place until almost doubled in size, about 45 minutes.

Turn the dough out onto a generously floured surface and roll into a 12-x-8-inch rectangle. Using a sharp knife or pizza cutter, cut the rectangle into 12 bars (4-inches long by 2-inches wide). Transfer to an oiled, waxed paper-lined baking sheet, cover, and let rise for 30 minutes.

Heat 1-1/2 to 2 inches of oil in a deep fryer or heavy skillet. Carefully fry 2-4 bars at a time for 1-2 minutes per side until dark golden. Transfer to a paper towel-lined baking sheet to drain.

Prepare the glaze by combining butter and brown sugar in a small saucepan with 3 tablespoons of milk. Whisk over medium heat for 3-5 minutes until sugar is dissolved. Remove from heat, stir in corn syrup and maple extract, then whisk in powdered sugar 1/2 cup at a time until smooth.

When the bars are cool enough to handle, dip the top side into the warm maple glaze. Set on a wire rack with the glaze side up until the glaze sets, about 10-15 minutes.