Preheat oven to 400°F and position two racks in the top and lower thirds of the oven. Line 2 baking sheets with parchment paper.
Place the butter, water and salt in a medium saucepan over low heat. Cook, stirring from time to time with the wooden spoon, so the butter melts evenly. When the butter has melted, increase the heat and bring to a boil. Immediately remove the pan from the heat and add the flour all at once. Beat vigorously with the wooden spoon until the dough comes together in a mass around the spoon. Place the pan back over medium heat and continue to cook, beating, for another minute or so to dry out the dough—the pan will have a thin film of dough on the bottom.
Immediately transfer the dough to the bowl of a stand mixer. Beat on medium speed for 1 minute to slightly cool the dough and develop the gluten. With the mixer on medium speed, add the eggs, one at a time, allowing each addition to blend completely into the dough before continuing. When all the eggs are incorporated, the mixture should be shiny and elastic and stick to the side of the bowl. It should also pass the “string test”: Place a bit of dough between your thumb and forefinger and pull them apart. The dough should form a stretchy string about 1 1/2 to 2 inches long. If the dough has not reached this stage, whisk another egg and continue adding it, a little at a time, until the dough is smooth. With the mixer on low, mix in paprika, pepper, cheese and thyme and mix until well combined.
Spoon the pâte â choux dough into a pastry bag fitted with a 1/2-inch plain round tip. Pipe the dough into 1-inch circles, about 1/2-inch high. (To stop the flow of dough from the pastry bag and disconnect it from the piped dough, slice a lightly oiled paring knife across the opening of the tip.)
Brush a light coating of egg over the tops of the piped dough with a silicone pastry brush, being careful that the egg does not drip down the sides (it will glue the gougères to the parchment).
Bake both sheets of the gougères for 15 minutes, then reduce the oven temperature to 350°F, switch the sheets between the racks, rotate the pans from front to back, and bake for 15 minutes longer. Reduce the temperature again, to 300°F, and bake 10 minutes longer (to dry out the interior). The gougères should be a deep golden brown, with no bubbling moisture visible around the sides. Transfer to a rack to cool slightly and serve warm or at room temperature.