Preheat Oven: Heat oven to 390°F (200°C).
Boil Butter Mixture: In a pot, mix butter, milk, water, salt, and sugar. Bring to a boil, ensuring the butter melts completely.
¼ cup milk, 3 tbsp unsalted butter, ¼ tsp salt, 1¼ tsp sugar, ¼ cup water
Add Flour: Off the heat, add flour all at once and stir. Return to low heat, stirring continuously for 2-3 minutes until the dough is thick, glossy, and leaves a thin film on the pot bottom.
2½ oz all-purpose flour
Cool Dough: Transfer dough to a bowl and let it cool for 10 minutes, spreading it out to cool faster.
Whisk Eggs: Beat eggs in a separate bowl.
2 large eggs
Melt Butter: Melt the butter for brushing the chouquettes.
1 tbsp unsalted butter
Mix Dough: With the dough slightly cooled, mix with a hand mixer. Gradually add half the eggs, mix, then add the remaining eggs bit by bit until the dough forms a 'V' shape when lifted with a whisk or spatula.
Prepare Tray: Line a baking tray with parchment paper or a silicone mat.
Pipe Dough: Fill a piping bag and pipe dough onto the tray in 1 1/4-inch mounds, leaving 1/2-inch space between each.
Brush: Using your fingers, brush all the dough mounds with the melted butter, patting down any peaks.
Sprinkle Sugar: Sprinkle the dough generously with pearl or sanding sugar.
1 oz Swedish pearl sugar
Bake: Place the baking tray in the oven, immediately reduce temperature to 356°F (180°C), and bake for 30 minutes until slightly browned. Open the oven door slightly for the last 5 minutes of baking.