Peppermint Cream-Puff Ring with Chocolate Glaze

Position rack in center of oven and preheat to 425 °F. Using 8-inch round cake pan bottom as guide, draw 8-inch circle on piece of parchment paper; turn paper over and place on baking sheet. Bring first 5 ingredients to boil in heavy medium saucepan over medium-high heat. Add flour all at once; mix in with wooden spoon. Reduce heat to medium-low. Stir constantly and vigorously until dough comes together in smooth mass. Continue cooking until dough pulls away from sides of pan, stirring constantly, about 2 minutes. Transfer hot dough to heavy-duty electric mixer fitted with paddle attachment. Add 3 eggs, 1 at a time, and beat until each egg is fully incorporated before adding next. Whisk remaining egg in small bowl to blend; very gradually add enough egg to form thick, shiny, and silky dough. Spoon warm dough into pastry bag fitted with 3/4-inch-wide plain piping tip. Pipe 1-inch-thick ring of dough along outline of circle on parchment. Pipe second ring of dough inside first ring, just touching first. Pipe third ring of dough on top of circle where first and second rings of dough meet. Bake dough ring 15 minutes. Reduce oven temperature to 375 °F and bake until ring is puffed, brown, and firm, about 20 minutes. Turn off oven; leave ring in oven 5 minutes. Using serrated knife, cut off top of pastry ring in 1 piece. Cool both top and base pieces completely. Using fingers, gently remove any soft dough from inside of base and discard. (Can be made 6 hours ahead; let stand at room temperature.)