In a medium bowl, whisk together the melted butter, milk, water, and vanilla extract. (You want the temperature of the ingredients to be warm, at least 105 degrees. If the ingredients are too cool the yeast will not activate. If necessary, put in the microwave for 10 seconds). Set aside.
In the bowl of a stand mixer, mix together the flour, sugar, salt, and yeast.
Add in the melted butter mixture. Then, add the eggs, one at a time.
Using the dough hook attachment, knead until the dough is soft and smooth, about 2 minutes. If it seems too dry add a little water, if it seems too sticky add just a little flour, up to a 1/4 cup.
Place the dough in a lightly greased bowl, cover, and allow to rest until doubled in size, about 1 hour.
Grease a 9" x 5" loaf pan. (You can also use a longer/large pan if needed, the bread will simply not be as compact.)
Gently deflate the dough and turn it out onto a lightly floured surface.
Roll the dough into a 12" x 20" rectangle.