Preheat oven to 400°F (200°C) on convection. Line 2 large baking sheets with parchment paper; set aside.
Gougère: In a medium saucepan, heat milk, water and butter over medium-high heat until butter is melted. Add flour and cook, stirring constantly with a wooden spoon for 2 minutes.
Scrape mixture into the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed for 1 minute. Reduce speed to medium-low; add one egg at a time, beating until fully incorporated before adding the next egg; about 6 minutes.
Using a spatula, fold 1 cup (250 ml) of the cheese, thyme and pepper into the mixture. Add mixture into a piping bag fitted with a large plain tip. Pipe about 1 tbsp (15 ml) mounds onto baking sheet; about 24 mounds on each baking sheet. Season extra pepper onto mounds and sprinkle with remaining cheese.
Bake for about 20 minutes or until golden brown. Repeat with remaining mixture on second baking sheet to make about 48 puffs total.
Strawberry Butter: In a small bowl, mash together butter and jam until fully combined.
Serve gougère warm with strawberry butter for spreading.
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