Preheat the oven to 375°F
Pour the milk and water into a small saucepan. Add the butter and place the pan over medium heat. Stirring constantly, simmer the mixture until the butter has melted completely.
Add the salt, sugar and flour. Stir to form a thick almost dough-like paste. Cook the paste for 2-3 minutes or until a film forms on the bottom of the pan and the dough starts to smell vaguely like brown butter.
Transfer the paste to a stand mixer fitted with a paddle attachment and mix on medium-low for 1 minute to cool the mixture off.
Add one of the eggs to the dough and mix until fully integrated. Scrape down the sides and add the next egg. Repeat in this manner until all the eggs are in and fully integrated.
Transfer the finished dough to a piping bag fitted with a star tip.
Line a baking sheet with parchment paper. Pipe a small dollop of the choux pastry on each corner of the parchment paper and use it stick the paper to the sheet.
Pipe the choux pastry onto the parchment paper in a series of straight lines. Slowly count to 5 to yourself to keep the eclairs relatively uniform in length. Wet your index finger in a bowl of cold water and smooth out the flicks and tails at the end of your eclairs.
Crack an egg into a small bowl and add a tablespoon of water. Whisk to combine. Brush the mixture lightly over the surface of your eclairs and transfer them to the oven.
Bake the eclairs for 25 minutes or until golden brown. Do not open the oven while they are baking, they may collapse. Take your finished eclairs out of the oven and set aside to cool.
Using a toothpick create two holes on either end of the base of each eclair. Using a piping bag fitted with a fine round tip, pipe the Lilac pastry cream into each hole until it starts to spill out. Repeat with all the remaining eclairs.