Preheat your oven to 400 degrees F. Line two baking sheets with silpats or parchment paper.
In a medium saucepan, combine the butter, milk and kosher salt and bring to a boil. Remove the pan from the heat and add the flour. Whisk for a few minutes, then return the pan to the heat and continue whisking to dry the paste out slightly. Remove the pan from the heat again, switch to a wooden spoon, and begin adding the eggs one at a time to fully incorporate into the dough before adding the next one. Your arm is going to get a workout here. Stir in the goat cheese and chopped rosemary.
You can either use a piping bag or drop tablespoonful of the dough onto the baking sheets.
Use the whisked egg to brush the tops of each gougère. Sprinkle each top with a little grated Parmesan cheese.
Place the baking sheets evenly spaced apart in the middle of your oven. Bake for 15 minutes, halfway through rotating each pan. Bake until gougères are golden brown and puffed. Serve immediately warm or at room temperature. Enjoy!