Preheat oven to 425°. Line a large rimmed baking sheet with parchment paper.
In a medium saucepan, bring ½ cup water, butter, salt, and nutmeg to a rolling boil over high heat. Add flour all at once, stirring with a spoon until blended. Cook over medium heat, stirring vigorously until mixture pulls away from sides of pan and forms a ball, approximately 1 minute. Remove from heat, and let stand for 1 to 2 minutes to cool slightly.
Transfer dough to a large bowl. Add 2 eggs to dough, one at a time, beating at medium speed with a mixer until dough is smooth and shiny. Add cheese, beating until well blended. Transfer dough to a large piping bag. Cut a ¾-inch opening in tip. Pipe 12 (¾-inch-wide) mounds onto prepared baking sheet.
In a small bowl, lightly beat remaining 1 egg. Brush tops of dough with egg, being careful not to allow egg to drip down sides.
Bake for 15 minutes. Reduce oven temperature to 325°. Bake until golden brown and very firm, 3 to 5 minutes more. Turn off oven. Pierce the side of each gougère with the tip of a knife. Return gougères to oven, and let stand with oven door ajar for 10 minutes. Remove from baking sheet, and let cool completely on a wire rack.
In a small bowl, stir together mayonnaise, chives, tarragon, mustard, lemon zest, and pepper. Add shrimp, gently stirring until combined. Cover and refrigerate until ready to serve.
Using a serrated bread knife, cut gougères horizontally in half. Place approximately 1 tablespoon shrimp filling on each bottom half. Cover with top halves of gougères. Serve immediately.
Notes
MAKE-AHEAD TIP: Gougères are best when made within 4 hours of serving, but shrimp salad can be prepared up to a day in advance, placed in a covered container, and refrigerated until serving time. Assemble just before serving.
Recipe by TeaTime Magazine at https://teatimemagazine.com/shrimp-filled-gougeres/