Preheat oven to 425 degrees F. Line two baking sheets with silicone baking mats or parchment paper.
In a medium saucepan, add the butter, milk, water, salt and pepper. Heat over medium-high heat until the mixture comes to a boil.
Add the flour, reduce the heat to medium-low, and stir with a heat-proof spatula until a smooth dough is formed, about two minutes.
Add the cooked dough to the bowl of a stand mixer fitted with the paddle attachment. Stir the dough on low speed until it stops steaming - about two minutes.
Add the eggs one at a time with the mixer on low speed and stir until each egg is completely incorporated into the dough.
Add the shredded cheese and mix with the mixer until it is completely incorporated into the dough.
Using a medium-sized cookie scoop, transfer the dough to the prepared baking sheets. You'll have 12 Gougeres on each pan.
Bake the Gougeres for 10 minutes. Reduce the oven temperature to 350 degrees F. Rotate the pans, top to bottom and front to back. Continue baking for an additional 10 minutes, until Gougeres have puffed up and are firm to the touch.
Allow to cool completely before removing the tops and adding egg salad or your choice of filling.