In a stand mixer, whisk together water and yeast. Let sit for 5 minutes.
Add egg, milk, sugar, vinegar, vanilla, salt, and cinnamon. Whisk on low until combined.
Switch to the dough hook attachment and add in flour. Knead on low speed until the dough is smooth and elastic. It should form a ball, but still be slightly sticky and soft. Mix in 1 to 2 tablespoons of flour at a time IF the dough is too wet to handle.
Last, add in the melted butter and mix on low until fully incorporated.
Cover the dough and let rise in a warm environment for 4 to 6 hours, or until doubled in size.
Heat 2 inches of canola oil over medium-high heat in a large skillet or dutch oven. Allow the oil to heat to 350 degrees Fahrenheit, this should take about 5 to 10 minutes.
Meanwhile, place dough on a floured surface and roll it out into a rectangle, about ¼ inch thick. Cut into 2-inch by 2-inch squares.
Carefully add the beignets to the oil, frying about 3 to 4 at a time. As soon as they rise to the oil surface and puff up, flip them over. Fry until golden brown on both sides, about 30 seconds to 1 minute. Transfer beignets to a wire rack to cool.