Step 1: Make the Dough

1. Activate the Yeast: In a small bowl, combine the warm milk, warm water, and a pinch of sugar. Sprinkle the yeast over the top and let it sit for 5-10 minutes until foamy.
2. Mix the Dough: In a large mixing bowl, combine the flour, sugar, and salt. Add the yeast mixture, 1 egg, and softened butter. Mix until a shaggy dough forms.
3. Knead the Dough: Turn the dough onto a lightly floured surface and knead for 8-10 minutes, or until smooth and elastic. Alternatively, use a stand mixer with a dough hook for 5-7 minutes.
4. First Rise: Place the dough in a lightly greased bowl, cover with a clean towel or plastic wrap, and let it rise in a warm place for 1 hour, or until doubled in size.

Step 2: Prepare the Cream Cheese Filling

1. In a medium bowl, beat the cream cheese, powdered sugar, and vanilla extract until smooth and creamy.
2. Transfer the filling to a piping bag fitted with a round tip or a zip-top bag with a corner cut off. Refrigerate until ready to use.

Step 3: Shape and Bake the Buns

1. Divide the Dough: Punch down the risen dough and divide it into 8 equal portions. Roll each portion into a smooth ball and place them on a parchment-lined baking sheet, spaced slightly apart.
2. Second Rise: Cover the buns with a towel and let them rise for 30 minutes.
3. Egg Wash and Bake: Preheat the oven to 350°F (175°C). Beat the remaining egg and brush it over the tops of the buns. Bake for 15-18 minutes, or until golden brown. Let the buns cool slightly.

Step 4: Fill the Buns

1. Cut the Buns: Using a serrated knife, make six evenly spaced cuts on each bun, slicing about two-thirds of the way through to create wedges. Be careful not to cut all the way through.
2. Pipe the Filling: Pipe the cream cheese mixture into the cuts of each bun, ensuring all wedges are filled generously.

Step 5: Coat with Garlic Butter Sauce

1. Make the Sauce: In a medium bowl, whisk together the melted butter, milk, mayonnaise, honey, minced garlic, parsley, and salt.
2. Dip the Buns: Holding each bun by the bottom, dip it into the garlic butter sauce, coating the top and sides thoroughly. Place the coated buns back on the baking sheet.
3. Bake Again: Return the buns to the oven and bake for an additional 5-7 minutes, or until the sauce is bubbly and slightly caramelized.

Step 6: Serve and Enjoy

1. Serve the buns warm, garnished with extra parsley if desired. These buns are best enjoyed fresh but can be reheated gently if needed.