Whisk together the flour, brown sugar, baking powder, and salt in a medium bowl. Whisk together the milk, egg, and vanilla in a small bowl. Set both bowls aside.
Heat 1 tablespoon of the butter in a large, heavy skillet over medium-low heat. Pan-fry the sausages until they are nicely browned and cooked through, 7 to 8 minutes. Place the sausages on a plate and wipe out the skillet.
Melt 2 tablespoons of the butter in the skillet over low heat. Pour the melted butter into the milk-egg mixture, whisking vigorously. Pour the wet ingredients into the flour mixture and whisk just to combine. It’s okay if there are some small lumps in the mixture.
Melt the remaining 1 tablespoon butter in the same skillet over medium heat. Working quickly, pour in 1/4 cup batter, coaxing it into a round shape with the back of a spoon. Repeat for as many 1/4 cupfuls as will fit comfortably in the pan. Cook, moderating the heat so that they don’t burn, until the bubbles that form around the edges of the pancakes pop, about 4 minutes. Flip and cook until the pancakes are slightly puffed in the center and golden brown on the bottom, 2 to 3 minutes longer. Transfer to a plate and repeat for the remaining batter.
To serve, wrap the sausages in the pancakes, and serve with maple syrup.