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Boil the Water and Butter: Combine the water, butter, salt, and mustard (if using) in a 2- to 4-quart saucepan and bring to a rolling boil.
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2. Add the Flour: Once all the butter has melted, remove the pan from heat and add the flour all at once. Stir vigorously until the mixture comes together and resembles mashed potatoes.
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3. Cook the Dough: Return the pan to medium-low heat and stir for 3-5 minutes to dry out the dough. The dough is ready when it glistens and is thick enough to hold a spoon upright. Some starch buildup on the bottom of the pan is normal.
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4. Add the Eggs: Transfer the dough to the bowl of a stand mixer fitted with a paddle attachment. (The following steps can also be done with a hand mixer or by hand with a stiff spatula.) Beat the dough on a medium-low speed for one minute until it stops steaming and is just warm to the touch.
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Continue beating and add the egg in four additions. Wait for each addition to be absorbed and for the dough to smooth out before adding the next. Scrape down the sides of the bowl if needed between additions. In the end, the dough should come together in a very smooth, creamy batter. Beat in the cheese.
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5. Bake the Gougeres: Heat the oven to 450u00b0F. Scoop rounded tablespoons of dough onto sheet pans lined with silicone mats or parchment paper. Space them at least an inch apart.
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Bake the gougeres for 5 minutes and then turn the heat down to 350u00b0F. Bake for another 20-25 minutes, rotating the pans once during baking. The finished gougeres will be puffed, deep golden-brown, and dry to the touch (the cheese may still be bubbling a bit). They will also feel light and hollow when picked up.
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6. Cool the Gougeres: Transfer the sheet pan to a cooling rack. Serve warm or room temperature. Leftovers can be stored in the refrigerator and re-crisped in a warm oven before serving or frozen for up to three months.
