Prepare the artichokes by trimming the top off each outside leaf with a pair of scissors. Cut about 2cm off the top of the artichoke, as well as the stem. Wash and then slice the artichoke in half and scrape out the fuzzy centre (known as the ‘choke’) with a spoon.
Boil the artichokes until tender, for around 20 minutes. Drain and pat dry with paper towels.
Snap off and discard the tough, bottom ends of the asparagus.
Whisk the olive oil, lemon juice, and salt and pepper together. Brush the cut sides of the artichoke halves and the asparagus with the olive oil mixture. Set aside some olive oil for later use. You can choose to grill the vegetables immediately or leave them to marinate.
Cut the lemon into wedges or slices and peel the cloves of garlic.
Warm the grill and grill the artichokes, cut side down, for about 10 minutes. Flip the artichoke over and grill for a further 5 minutes.
Grill the asparagus for 1-2 minutes, turning often. Add the lemon and garlic to the grill and remove once they are golden brown in colour.
Place the artichokes and asparagus on a plate and drizzle with reserved olive oil mixture. Garnish with the lemon parts.