In the bowl of a stand mixer, stir together 2 cups (9 ounces) of the flour, the sugar, yeast, and salt.
In a small saucepan, heat the milk and butter over low heat just until the butter is melted. Remove from the heat, add the water, and set aside until warm (120 to 130°F [49 to 54°C]), about 1 minute. Add the vanilla extract.
Pour the milk mixture over the flour-yeast mixture and, using a rubber spatula, mix until the dry ingredients are evenly moistened. Attach the bowl to the mixer, and fit the mixer with the paddle attachment. With the mixer on low speed, add the eggs, 1 at a time, mixing just until incorporated after each addition. Stop the mixer, add 1/2 cup (2 1/4 ounces) flour, and resume mixing on low speed until the dough is smooth, 30 to 45 seconds. Add 2 more tablespoons flour and mix on medium speed until the dough is smooth, soft, and slightly sticky, about 45 seconds.
Sprinkle a work surface with 1 tablespoon flour and turn the dough onto the flour. Knead gently until the dough is smooth and no longer sticky, about 1 minute, adding an additional 1 to 2 tablespoons flour only if the dough is unworkably sticky. Place the dough in a large bowl, cover the bowl tightly with plastic wrap, and let the dough rise in a warm place (about 70°F [21°C]) until doubled in size, 45 to 60 minutes. Press the dough gently with a fingertip. If the indentation remains, the dough is ready for the next step.