Position the racks in the upper and lower third of the oven. Heat the oven to 400ºF (200ºC). Line two baking sheets with parchment paper or silpat baking mats.
In a medium saucepan over medium-high heat bring the milk, water, butter and salt to a rapid boil.
Add the flour, lower the heat to medium-low and immediately stir with a wooden spoon while the dough comes together.
Continue to cook, stirring constantly, to dry out the dough, until the dough becomes smooth ball that easily pulls away from the sides of the pan, an additional 1 to 2 minutes. Remove from heat.
Add the dough to the bowl of a stand mixer fitted with a paddle attachment. Turn the speed to low and add the eggs in one at a time, beating well after each addition. Once all the eggs have been added, the dough will be shiny, thick and fall from a spoon.
Add the cheese, cayenne and lobster meat and beat to combine.
Drop the choux pastry dough in rounded tablespoons onto the baking sheets two inches apart.
Brush each with the egg wash mixture then bake for 15 minutes. Rotate the pans and continue to bake until the gougères are crisp on the outside and slightly soft on the inside, about an additional 15 minutes.
Top with chives and serve warm. The gougères can be made ahead of time, frozen and reheated in the oven.