Microwave milk for about 30 seconds or until it's warm (100-110 degrees F). Add 1 teaspoon sugar and 2 ¼ teaspoon yeast. Let sit for 5 minutes or until the yeast is activated and the liquid is foamy.
Combine 3 cups flour with ½ teaspoon salt and the remaining sugar in the bowl of a stand mixer. Add the yeast mixture, ¼ cup softened butter, and 1 teaspoon vanilla and mix until the dough starts to come together. Add the eggs one at a time until well combined. Knead the dough for 5-7 minutes or until the dough is smooth, tacky, and elastic. Transfer the dough to a clean bowl, cover with plastic wrap and let proof for 1 hour or until doubled in size.
Prepare the filling. Combine ½ cup granulated sugar with ¼ cup meyer lemon zest.
Punch down the dough. Dust a clean work counter with flour and transfer the dough to the counter. Roll it into an 18x15 inch rectangle. Brush on the melted butter and sprinkle the sugar lemon zest mixture on top in one even layer. Cut the rectangle into 5 strips lengthwise (each strip should be 18x3 inches). Stack the strips on top of each other, placing the last strip with the filling face down. Cut the stack into 6 even squares. Carefully place each stack into a muffin pan. Cover and let rise 30 minutes.
Preheat oven to 350 degrees F.
Bake pull apart bread for 18-20 minutes or until golden brown. Remove from the oven and let cool on a wire rack.
Prepare the icing. Whisk together the powdered sugar with Meyer lemon juice. Drizzle icing over the bread and serve while warm.