Preheat oil to 350˚F.
For white queso dip: Melt butter in a small saucepan, over medium heat. Add flour and whisk together for 2 to 3 minutes.
Whisk milk into the roux and continue to whisk together until the mixture is smooth. Simmer, stirring occasionally, until mixture is thick enough to coat the back of a wooden spoon, 3 to 4 minutes. Lightly season with salt.
Remove from heat and stir in shredded cheese, a small handful at a time, waiting until the cheese melts to add more.
Continue to add cheese until all has been added and dip is smooth. Adjust seasonings. Set aside and keep warm (stirring 1 tablespoon hot milk, at a time, if dip gets too thick).
For churros: Combine flour, chives, thyme, baking powder, and salt in a mixing bowl and whisk together.
Using a wooden spoon and stir in the boiling water. Continue to stir until the steam dissipates and thick dough forms.
Gently fold in ¼ cup Parmesan until just combined.
Transfer half of the dough into a piping bag fitted with a small, star tip.
Holding the piping bag over the hot oil, carefully squeeze 2 inches dough from the piping bag and cut the end off with a pair of kitchen scissors.
Repeat until you have several churros frying, making sure not to overcrowd the oil.
Fry churros for 4 to 5 minutes or until golden brown and crisp. Transfer to a tray lined with paper towels and season with salt.
Repeat steps 6 to 9 until all of the churros have been fried.
Transfer churros to a large platter, top with remaining Parmesan and serve with queso dip.