Place 1 1/4 cups water, butter, brown sugar and salt in a saucepan and bring to a boil. Add the flour all at once and stir vigorously until mixture forms a ball and smells like toasted flour. Transfer to the bowl of a mixer and paddle the dough while adding the eggs one at a time, allowing machine to incorporate each egg before the next addition. Scrape down the sides and bottom of the bowl often, and then add the vanilla. Lastly, add the dulce de leche and mix until thoroughly incorporated. Transfer the mixture to a container and refrigerate until ready to fry.
Heat vegetable oil for frying in a pot. Place the churro batter in a pastry bag fitted with a large star tube. When the oil is 375 degrees F, drop 2-inch-long churros from the pastry bag into the oil and fry until well browned and floating, about 4 minutes. Centers will still be a tiny bit gooey because of the dulce de leche in the batter. Remove the churros to a paper-lined plate. Combine the granulated sugar and cinnamon and toss with the hot churros.
To serve, place the churro tots on a small plate or in a shallow bowl accompanied by a dollop of Whipped Cream for dipping.
Dulce de Leche:
Combine the milk, sugar and baking soda in a large, heavy saucepan. Place over medium heat and cook, stirring often, until the mixture becomes a caramel color and is thick enough that you can see the bottom of the pan as you stir, 45 minutes to an hour. By volume it should be reduced to about 1 cup. Cool to room temperature and use immediately, or store in refrigerator until ready to use.
Combine the cream and sugar in a cold mixing bowl and whip until stiff peaks form.