Preheat the oven to 200C, fan-forced. Halve the leek lengthwise and rinse under running water to remove any hidden dirt. Thinly slice the leek, discarding any dark green stem. Crush or finely chop the garlic. Quarter the carrot and zucchini lengthwise, then thinly slice. Drain and rinse the beans.
Step 2 — Cook leek
Heat 1 tbs olive oil in a medium saucepan over medium-high heat. Cook the leek and garlic, stirring, for 2-3 mins until softened.
Step 3 — Prep tomato stock
Meanwhile, crumble the stock cubes into a heatproof jug, add half the tomato paste, 750ml (3 cups) boiling water and 1 tsp sugar and stir to dissolve.
Step 4 — Add tomato and carrot
Add the tomatoes, carrot and tomato stock to the leek mixture. Increase the heat to high and bring to the boil. Reduce the heat to medium and cook for 12 mins.
Step 5 — Make parmesan crisps
While the soup is cooking, line 2 oven trays with baking paper. Finely grate the parmesan directly onto the lined trays in a thin layer (this gives a better result than scattering the parmesan by hand). Bake for 4-6 mins until golden. Cool, then break into large pieces.
Step 6 — Cook vegetables and serve
Add the beans, zucchini and peas to the soup and cook for a further 3 mins or until tender. Remove the pan from the heat. Taste, then season with salt and pepper. Divide the minestrone among bowls. Drizzle with the pesto and serve with the parmesan crisps.