Preheat the oven to 220C/gas 7, and insert two large baking trays or a pizza stone to warm up.
Place a large pan of salted water over a high heat and, once boiling, cook the new potatoes for 15–20 minutes, until just tender. Drain and leave to steam dry in the colander, then transfer to a bowl and lightly crush them with a fork.
Meanwhile, make your pesto. Using a pestle and mortar or a food processor, crush the garlic with a generous pinch of salt, then add the basil leaves and grind to a paste. You may need to do this in a few stages, as it won’t all fit in one go.
Add the pine nuts and bash well, then stir in the parmesan and 6 tablespoons of olive oil to loosen. Mix in a good pinch of pepper, and leave to one side.
Cut the dough in half and, on a floured surface, roll out both pieces to around 5mm thick and 30cm in diameter. Dust the hot trays or pizza stone with semolina, then transfer the bases to the trays.
Divide the pesto between the bases and spread to coat. Toss the potatoes with the rest of the olive oil, season, then scatter them on top of the pizzas. Tear up the mozzarella and poke it between the potato, then throw on a few basil leaves.
Pop the pizzas in the oven for 15–18 minutes, or until just golden and bubbling. Dress the pea shoots and peas with a squeeze of lemon juice, scatter them over the cooked pizza, and serve.