Preheat the oven to 400 degrees. On the reverse side of a sheet of parchment paper, draw out ten 5" lines with space between them, to serve as a guide when piping the eclairs. Line a large baking sheet with the parchment.
Sift the 1 1/4 cups of All-Purpose flour into a bowl.
In a medium saucepan, over medium heat, using a wooden spoon or sturdy rubber spatula, add the 1/2 cup of butter and stir until melted. Once the butter is melted add the 1/4 tsp salt, 2 tsp of sugar, 1/2 cup whole milk, and 1/2 cup of water. Bring to a boil. As soon as it starts to boil remove it from the heat.
All at once, pour the sifted flour into the saucepan and stir vigorously to quickly incorporate the flour and form a non-lumpy dough. Move the saucepan back to the burner and cook over medium heat for 1 minute.
After this, transfer the choux dough to your Kitchenaid mixer fitted with the paddle attachment. Add each egg to the dough one at a time, mix on medium speed until each one is fully incorporated. After the first two eggs, scrape the sides of the bowl with a spatula to make sure the dough is mixing thoroughly. You may not need the 5th egg. The Choux dough should not be too runny! It's ready to be piped when you lift the paddle attachment out and the dough slowly forms a V dropping off of the paddle.
When the Choux dough is the right consistency, scoop it all into a 16" piping bag. Once all the dough is in the piping bag, measure 2 1/2" up from the bottom tip of the bag and cut it off. This will form the perfect size opening to create 10 nice-sized eclairs.
Follow the 5" guides you drew earlier and pipe the 10 eclairs. You can use a plastic bench scraper to keep pushing the Choux dough down in the piping bag to avoid getting air bubbles in the dough while piping.
Once the eclairs are all piped, place them on the middle rack of the oven and bake at 400 degrees for 10 minutes. After this, turn the oven down to 375 degrees and bake for an additional 30-45 minutes. They will be very golden brown when they are finished baking, don't be tempted to remove them early, or open the oven as they're baking, as they will go flat.
Once they are finished baking, turn off the oven and crack the door open, allow them to sit like that for 10 minutes, and then remove them from the oven and finish cooling them on a cooling rack.