In a deep skillet, heat 2 inches of vegetable oil to 375°F over medium-high heat.
In a medium saucepan, combine 1- cup water, brown sugar, salt, and butter and bring to a boil, 10 to 12 minutes. Remove from the heat and add flour, stirring with large wooden spoon until well blended. In a bowl, mix eggs and vanilla. Stir into flour mixture until completely incorporated.
In a large re-sealable plastic bag or a paper bag, combine granulated sugar and cinnamon and set aside.
Spoon dough into a pastry bag fitted with a large star tip. Carefully squeeze dough into hot oil in 4-inch strips, being careful not to splash oil. Fry dough in small batches (about 6 at a time) until golden brown, 1 to 2 minutes per side. Drain churros on paper towel–lined plate. Add warm churros to cinnamon sugar and shake to cover. Serve warm.