Combine 1/3 cup sugar and cinnamon in a shallow pan; set aside for coating.

In a medium bowl, cream together butter and remaining 1/3 cup sugar using an electric mixer until creamy.

Add eggs and vanilla extract, blending well.

Mix in 1 cup of flour, baking powder, and salt; blend well, then mix in the milk.

Add the remaining flour and mix to form a soft dough.

Turn the dough out onto a floured surface and knead for 1-2 minutes until smooth, adding more flour if sticky.

Roll out the dough to 1/4-inch thickness, flouring the surface and rolling pin as needed to prevent sticking.

Cut the dough into 2-inch rounds using a cookie cutter.

In a heavy skillet or Dutch oven, heat 2 inches of vegetable oil over medium heat to 350 degrees.

Fry six 2-inch rounds at a time for 1-1/2 to 2 minutes until puffy and golden, turning often with tongs.

Drain the fried doughnuts on paper towels.

Toss the hot doughnuts in the sugar and cinnamon mixture to coat.