Thaw the puff pastry at room temperature. Peel the garlic. Wash the basil and dry on paper towels. Grate Parmesan on a medium grater. Peel the pine nuts if necessary. Prepare the blender. Line a baking tray with baking paper.
Make pesto. Place garlic cloves, basil leaves, pine nut kernels into a blender bowl, pour in half the oil, beat well. Add the grated cheese and remaining butter, mix the ingredients and season with salt and pepper.
Form a baguette. Dust your work surface with flour and roll out the puff pastry. Brush on three sides at a distance of 3-4 cm from the edge with a mixture of yolk and milk. Don't touch one long side. Spread the filling onto the dough, starting from the ungreased edge. Roll up, pinch and fold the edges. Let the dough proof for 40-50 minutes.
Bake a baguette. Preheat the oven to 180 degrees in advance. Place the baguette on a baking sheet, brush with the egg-milk mixture, and make shallow cuts diagonally. Place in the oven for 30-35 minutes.
Cut the baguette with pesto sauce into portions. Serve with a cup of hot tea for breakfast.