1. Combine water, butter, brown sugar, and salt in a saucepan over medium heat and bring to a simmer.
2. Add flour all at once, stirring until a sticky dough forms and a starchy film coats the pan, about 5 minutes.
3. Continue cooking for 1 more minute, scraping the pan's bottom.
4. Transfer dough to a bowl, cool for 10 minutes.
5. Mix in vanilla bean paste and eggs, one at a time, until a sticky dough forms.
6. Preheat oven to 425°F and line baking sheets with parchment paper.
7. Pipe 3 to 4 inch dough ropes onto sheets using a pastry bag with a star tip.
8. Spray churros with cooking spray and water.
9. Bake until puffed and browned, 20 to 25 minutes. Vent heat briefly, leave oven door ajar, and let churros rest in oven for 10 minutes.
10. Remove churros, brush with melted butter, and toss in cinnamon sugar.