Preheat the oven to 350°F convection or 375°F conventional. Measure and separate the flour, salt, and sugar into one container and the eggs into another.

Combine the butter and milk in a high-sided, heavy-gauge saucepan. Heat on high until the butter has melted and the milk just reaches a boil simultaneously. Do not allow the milk to continue boiling.

Add the flour mixture all at once and stir to incorporate with a wooden spoon. Once the mixture thickens, stir vigorously to smooth out any lumps.

Continue cooking on high heat, stirring constantly, until a cohesive dough (panade) forms, pulls away from the sides of the pot, and leaves a visible film on the bottom of the pan, about 2 minutes.

Transfer the dough to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium-high to release steam.

Once the mixture stops steaming, add the eggs one at a time on medium speed, beating fully after each addition. Test consistency using the trench test: drag a finger through the dough and the channel should close slowly and steadily. If needed, beat a sixth egg separately and add in stages, stopping at no more than half an additional egg.

Adhere parchment paper to the baking sheet by squeezing small dots of choux paste at the corners and center, then pressing the parchment on top.

Pipe 2-centimeter rounds, holding the tip about 1 centimeter above the parchment and applying firm, consistent pressure. Twist the wrist across the top of each ball to detach cleanly. Brush the tops with beaten egg and reshape any points with the brush.

Bake at 350°F convection or 375°F conventional for 18 to 20 minutes for 2-centimeter rounds, rotating the trays at 10 minutes. Larger rounds (up to 4 centimeters) will take 20 to 25 minutes. Gougères are done when puffed, evenly golden brown, and feel light when lifted.