1. In a saucepan, combine water, butter, sugar, and salt. Bring just to a simmer until butter melts and mixture gently bubbles at the edges. Remove from heat.
2. Immediately add all the flour at once and stir vigorously until a smooth dough forms and pulls away from the sides of the pan.
3. Transfer the dough to a bowl and allow to cool for about 5 minutes until it is warm but no longer hot.
4. Add eggs one at a time, beating thoroughly after each addition until the dough becomes thick, glossy, and smooth, showing a slight V-shape when dropped from a spoon.
5. For the filling, beat the softened cream cheese until smooth and creamy. Add powdered sugar and mix until lump-free. Gently fold in cookie dough pieces. Chill the filling for at least 30 minutes to firm up.
6. Heat oil in a deep pot or fryer to 350°F (175°C), maintaining at least 2 inches of oil.
7. Mix the cinnamon and sugar in a shallow dish for coating.
8. Fit a large star tip on a piping bag and fill it with churro dough. Pipe 4-inch lengths directly into the hot oil, cutting the dough with scissors as you go.
9. Fry churros for about 2-3 minutes per side until golden and crisp, working in batches. Drain on paper towels and let cool for 3-5 minutes until warm but not hot.
10. Using a small piping tip and a separate bag, pierce one end of each churro and pipe in the chilled cheesecake filling.
11. Immediately roll filled churros in cinnamon sugar, then arrange on a serving plate and drizzle with warm chocolate sauce if desired.
12. Serve immediately for the best texture.