In a medium saucepan over medium heat, add the blueberries, sugar, water, lemon juice, and salt. Mix until combined. Cook over medium heat until the blueberries have softened and released their juices (5-8 minutes).
Mix together the cornstarch and water in a small bowl to create a slurry. Pour the slurry into the blueberry mixture, stir, and continue to cook until the mixture thickens (1-2 more minutes).
Remove from the heat and set aside to cool while you make the churro bites.
In a medium saucepan over medium-high heat, add the butter. Melt and continuously stir the butter until it begins to brown. Once it smells nutty and begins to brown, add the water, sugar, and salt. Mix and bring the mixture o a small rolling boil, while you stir occasionally.
Once the sugar is dissolved and the mixture starts to boll, turn the heat to low and add the flour. IMMEDIATELY and vigorously stir with a spatula or wooden spoon for 1-2 minutes, until the flour is fully incorporated. The dough will become a thick, smooth paste, leaving a slight film on the bottom of the pan.
Remove from the heat and scrape the dough into a large bowl or the bowl of a stand mixer. Allow to cool for 5-10 minutes (it needs to be cooled enough so when you add the eggs, the dough won’t cook them).
While cooling, prepare the cinnamon sugar mixture by mixing together the sugar and cinnamon in a small bowl. Set aside.
Using an electric hand mixer (or mix by hand with a spatula) or a stand mixer with the paddle attachment, beat the egg into the dough until completely incorporated. The dough should become smooth, glossy, and sticky.
Transfer the dough to a piping bag fitted with a large star tip.
Fill a large, heavy-bottomed pot or Dutch oven with oil, filling ~2” in the pan. Heat on medium-high until it reaches 360F-375F.
Line a wire cooling rack with a thick layer of paper towels.
While the oil is heating, cut out medium-sized squares or circles of parchment paper. These will be used for the churro bites to sit and make it easier to transfer to the oil once ready to fry.
Pipe 6-7, little 1-inch churro bites onto each piece of parchment paper, making any shape desired. You can alternatively pipe the dough over the oil and use clean kitchen scissors to snip off the dough every 1 inch; I prefer the parchment paper method!
Once the oil is hot enough, place 2-3 pieces of the parchment paper with the churro bites on them slowly into the oil. After 10-15 seconds, pull the paper away from the churro bites with metal tongs (it should almost immediately release).
Fry the churro bites for about 2-4 minutes (~1-2 minutes on each side), until crispy golden brown. I used a fork to continuously toss and turn them as they were frying!
Using a slotted spoon, carefully remove the churro bites from the hot oil and set on the rack with paper towels. Then, quickly toss and coat the churro bites in the cinnamon sugar mixture.
Serve the churro bites warm with the blueberry dipping sauce.
