Preheat oven to 375°F (190°C). Draw 18 (1½-inch) circles at least 1½ inches apart on each of 3 sheets of parchment paper. Place parchment, ink side down, on 3 rimmed baking sheets.

In a medium saucepan, bring butter, ½ cup (120 grams) water, milk, sugar, and salt to a rolling boil over medium-high heat. Stir in flour and herbs or spices with a wooden spoon until well combined. Cook, stirring constantly, until mixture pulls away from sides of pan and forms a ball, leaving a thin layer of residue on bottom of pan, 1 to 2 minutes.

Transfer mixture to the bowl of a stand mixer fitted with the paddle attachment; beat at medium-low speed until dough cools slightly and is no longer steaming, 2 to 3 minutes. Add eggs, one at a time, beating until combined after each addition. (Batter will be shiny and thick.) Fold in 4 ounces (113 grams) cheese.

Transfer batter to a large pastry bag fitted with a medium round piping tip (Ateco #806). Holding bag perpendicular to parchment, slowly pipe rounds into drawn circles on parchment. Using a wet finger, smooth tops if necessary. Sprinkle remaining 2 ounces (57 grams) cheese onto dough.

Bake, one pan at a time, for 10 minutes. Rotate pan, and bake until puffed and golden brown, 5 to 8 minutes more. (Let remaining pan stand at room temperature until ready to bake.) Let cool completely on pans. Refrigerate in an airtight container for up to 3 days.