In a small bowl, combine the warm water, warm milk, and sugar. Stir until the sugar dissolves. Sprinkle the yeast over the mixture, stir, and let it sit for about 5-10 minutes, or until it becomes frothy.
In a large mixing bowl, combine the flour and salt. Make a well in the center and add the yeast mixture, eggs, and vegetable oil. Mix the ingredients until a dough forms.
Knead the dough on a floured surface for about 8-10 minutes, or until it becomes smooth and elastic. Place the dough in a lightly oiled bowl, cover it with a clean kitchen towel, and let it rise in a warm place for 1-2 hours or until it doubles in size.
Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
Once the dough has risen, punch it down and divide it into smaller portions. Roll each portion into a long, thin rope shape, about 12 inches (30 cm) long. Place the ropes on the prepared baking sheet, leaving space between each one.
Bake the fartons for about 12-15 minutes or until they are golden brown.
While the fartons are baking, prepare the glaze. In a small bowl, whisk together the powdered sugar, warm water, and vanilla extract until smooth.
Once the fartons are done baking, let them cool for a few minutes on the baking sheet. Then, brush the glaze over the warm fartons.
Allow the fartons to cool completely before serving.