01 - In a large bowl, mix together flour, sugar, yeast, and salt until evenly distributed.
02 - Whisk warm milk, melted butter, eggs, and vanilla extract in a separate bowl until combined.
03 - Gradually add wet ingredients to dry and stir until a sticky dough forms.
04 - Knead dough on a floured surface for 5 to 8 minutes until smooth and elastic.
05 - Place dough in a greased bowl, cover, and let rise in a warm area until doubled in size, about 1 hour.
06 - Roll dough to 1/2-inch thickness and cut donuts using a 3-inch cutter and 1-inch cutter for centers.
07 - Arrange donuts and holes on parchment-lined trays, cover, and allow to rise for 30 minutes.
08 - Heat vegetable oil in a deep pot or fryer to 350°F (175°C).
09 - Fry donuts in batches for 1 to 2 minutes per side until golden brown; fry donut holes for 1 minute, turning once.
10 - Remove fried pieces and drain on a paper towel-lined rack to remove excess oil.
11 - Brush warm donuts with melted butter, then toss in a cinnamon sugar mixture composed of sugar and ground cinnamon.
12 - Serve immediately while warm for optimal flavor and texture.