In the bowl of a stand mixer, combine the bread flour, milk, and sourdough starter until a rough, shaggy dough forms. Cover the bowl and let it rest in a warm, draft-free spot for 1 hour.

Fit the mixer with the dough hook and knead on medium-low for 5 minutes. Add the salt and superfine sugar, then add the softened butter a few pieces at a time while the mixer runs. Continue kneading until the butter is fully worked in and the dough feels soft, smooth, and elastic, about 7 to 9 minutes more.

Transfer the dough to a lightly oiled bowl, cover, and let it rest for 30 minutes. Perform one stretch and fold by lifting one side of the dough and folding it over itself. Cover and rest another 30 minutes. Repeat this stretch and fold three more times, waiting 30 minutes between each round.

After the final stretch and fold, cover the bowl and let the dough proof for 2 to 3 hours, until it has grown by about 50%. Refrigerate the covered dough overnight to finish proofing.

Remove the dough from the refrigerator and let it sit at room temperature for about 30 minutes. Turn it out onto a lightly floured work surface and gently press or roll it into a 12 by 9 1/2 inch (30 x 24 cm) rectangle, about 1/8 inch (3 mm) thick. Spread the pesto evenly over the dough and arrange the cheese slices across the surface, then roll it up starting from one of the shorter sides.

Cut the rolled dough into about 1 1/4 inch (3 cm) thick slices. Nestle the slices cut-side up in a generously buttered or parchment-lined baking pan. Cover with lightly oiled plastic wrap and let the rolls rise for 2 to 3 hours, until puffy and nearly doubled.

Preheat the oven to 410°F (210°C) with convection if available. Bake the pull-apart rolls for 25 to 30 minutes, until deep golden brown. Let the bread cool in the pan for 1 hour before serving.