In a bowl of a stand mixer or a large bowl, combine the milk, yeast, honey, eggs, 3 ½ cups flour, and salt. Using the dough hook or your hands, mix for about 4-5 minutes until the flour is completely incorporated. Add 2 tablespoons of room temperature butter, mix for about 3 minutes until combined. If the dough is still sticky, add the remaining ½ cup of flour. The dough should be smooth and tacky, but not sticky.
Cover the bowl with plastic wrap and let sit in a warm area for 1 hour or until doubled in size.
Meanwhile, make the pesto. Wash and dry the basil leaves. In a food processor, combine basil, parmesan, pine nuts, garlic cloves, lemon juice, olive oil, salt, and pepper. Process until smooth.
Once the dough has risen, place it on a lightly floured work surface. Roll into a large rectangle about 12x18 inches. Spread the pesto evenly over the dough. Starting with the long edge closest to you, roll the dough into a log. When you reach the end, pinch along the edge to seal the log.
Using a sharp knife, cut the dough in half lengthwise, leaving a small portion at the end intact. Turn the halves to expose the filling. Cross the halves over each other, repeating this step until you have a long twisted dough.
Starting at the dough's top, coil the end towards the center like a snake, stopping halfway. Then at the bottom of the dough, coil the end towards the center in the opposite direction. The end dough shape should resemble figure 8. Carefully transfer to a parchment-lined baking sheet. Cover and let rise for 45 minutes.
Preheat the oven to 350F. Transfer the bread to the oven and bake for 30-35 minutes, until golden brown. Brush the top of the loaf with melted butter. Slice and serve warm smeared with additional butter.