Special equipment: a large pastry bag and 1/2-inch wide star tip
For the churros: Adjust the oven racks to the top and bottom thirds of the oven and preheat to 425 degrees F. Line 2 baking sheets with parchment.
Bring 4 tablespoons of the butter, 1 tablespoon of the sugar, the salt and 1/2 cup of water to a boil in a small saucepan over medium heat. Add the flour and stir constantly until the mixture forms a ball. Transfer to a large bowl and beat with an electric mixer on medium-low until the dough is cool to the touch. Add the eggs one at a time and beat until incorporated. Transfer to a large pastry bag fitted with a 1/2-inch wide star tip.
Pipe the batter into 2-inch lengths about 2 inches apart on the prepared baking sheets. Bake until golden brown, rotating the trays from top to bottom about halfway through, 15 to 18 minutes. Let sit on the baking sheets for a few minutes.
Melt the remaining 3 tablespoons of butter in a medium microwave-safe bowl. Toss 1/4 cup sugar and 1 teaspoon cinnamon together in another medium bowl. Working in small batches, dip the churros completely in the melted butter and let any excess drip off, then toss in the cinnamon-sugar until completely coated. Once the cinnamon-sugar mixture is too wet to coat, mix together a fresh batch with the remaining 1/4 cup sugar and 1 teaspoon cinnamon. The churros can be baked and coated 1 day ahead of time and stored at room temperature in an airtight container.
For the caramel-mango sauce: Bring, the sugar, salt and 1/4 cup of water to a boil in a small saucepan over medium heat. Continue to cook, swirling the saucepan occasionally but not stirring, until the mixture turns a medium caramel color (slightly darker than honey), about 5 minutes. Remove from the heat and stir in the mango and butter (take care, the mixture will bubble). Let the mixture cool for a few minutes in the saucepan, then transfer to a food processor and process until smooth. Keep refrigerated until ready to use. The caramel-mango sauce can be made a day ahead. Serve the churros with the caramel-mango sauce for dipping.
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