Heat your oven to 200°C/Fan 180°C/Gas 6. Fill a large saucepan with water and bring to the boil.
Tip the flour into a mixing bowl, saving a good pinch for rolling. Finely grate in half of the cheddar. Chop in the cold butter. Use your fingers to rub the ingredients together until they resemble coarse breadcrumbs.
Crack the egg over a small bowl and carefully separate the yolk from the white. Stir the yolk into the flour mixture with 3-5 tbsp cold water, until you have a soft dough. Wrap in cling film and put in the fridge to rest for 15 mins. Keep the egg white for later.
Scrub the potatoes and slice them thinly. Trim and finely slice the green garlic bulb. Tip both into the large pan and simmer for 3-4 mins till the potatoes are slightly softened but still holding their shape. Drain well, then rinse with cold water until cool. Shake well to dry, then pop in a large bowl.
Finely slice the sage and zest the lemon. Grate the remaining cheddar. Add all 3 ingredients to the potatoes and garlic, along with a good pinch of seasoning. Toss to mix everything together.
Lay the rested pastry on a sheet of greaseproof paper. Roll out into a rough circle a little larger than a dinner plate, using the reserved flour if needed. Slide onto a flat baking tray.
Layer the potatoes on top of the pastry, making sure to tuck in the other flavourings here and there. Leave a gap of 3cm around the sides. Season with salt and pepper. Fold up the pastry edges to create free standing sides. Brush the pastry with the remaining egg white.
Slide into the oven and bake for 40 mins. Snap the woody ends off the asparagus and slice the tender spears in half lengthways. Toss with 1 tsp olive oil. Remove the galette from the oven and scatter the asparagus over the top and sides. Return to the oven for 5-6 mins till just tender.
Squeeze the lemon juice into a small bowl and stir in 1 tbsp olive oil and a pinch of salt and pepper. Roughly tear the lettuce leaves into a bowl and spoon over the dressing. Serve along thick slices of the golden cooked galette.