Preheat oven to 350 degrees F and line a large baking sheet with parchment paper. Add the 1/2 cup butter, milk, 1/4 cup sugar and salt to a medium saucepan. Bring to a boil over medium-high heat. Add the flour and (using a heavy or wooden spoon) stir until combined.
Remove from the heat and add the eggs and vanilla. Stir quickly (so eggs don’t cook) until combined. At this point the mixture should resemble mashed potatoes. If it is too runny, add flour 1/4 cup at a time.
Spoon into a piping or ziploc bag and use a star piping tip to pipe dough (approximate 1-inch thickness and any length you desire) onto the prepared baking sheet. Baked in the preheated oven for 20 minutes, or until churros are golden brown.
While the churros are baking, melt the 1/2 cup butter in a small bowl and in a separate bowl combine the 1/2 cup sugar and cinnamon. Mix well. Remove the churros from the oven and immediately brush with the melted butter and coat with the sugar-cinnamon mixture (you can do this by sprinkling it over the tops and rolling or by shaking together in a ziploc bag). Enjoy immediately!
Notes
Use a heavy-duty or wooden spoon to stir in the flour and eggs. The mixture should be thick and you'll want to stir in the eggs as quickly as possible so they don't turn into scrambled eggs.
After the flour and eggs are incorporated, the mixture should resemble a thick bowl of mashed potatoes. If it seems too runny, add extra flour.
After the melted butter has been brushed onto the churros, you can coat them with the sugar-cinnamon mixture by sprinkling generously (and then rolling) or by shaking them together in a ziploc bag.