Roll out pastry to 1/8-inch thick.
Cut out an 8-inch pastry circle; transfer to a parchment paper-lined sheet pan, and prick all over with a fork.
Cover and refrigerate.
Heat oven to 425°F.
Bring 1/2 tablespoon sugar, 1/4 cup milk, butter, salt, and 1/4 cup water to a boil in a small pot.
Add 1/2 cup flour and cook, stirring, until it forms into a dough ball, about 20 seconds.
Transfer dough to a bowl.
Add whole eggs, one at time, while beating with an electric mixer; continue beating until dough is smooth and holds a soft peak.
Transfer dough to a pastry bag fitted with a 3/8-inch round tip and pipe 3 circles onto the chilled pastry circle: one along the outer edge, one around the middle, and one in the center.
On the same baking sheet, arranged around the pastry circle, pipe remaining dough into thirteen 1-inch-wide puffs.
Press down the pointy tops with a moist fingertip.
Bake pastry circle and puffs until beginning to brown, about 12 minutes.
Reduce heat to 325°F; bake until dry and golden – 6 minutes for puffs and 10 minutes for pastry.
Let the pastry circle and puffs cool.