How to Make It
Fill a medium bowl with water and squeeze the lemon half into it. Working with 1 artichoke at a time, snap off the dark outer leaves until you reach the yellow leaves. Cut off the top two-thirds of the leaves, leaving 1 1/2 inches at the base. Cut off the stem and peel the artichoke bottom, then scoop out the hairy choke with a melon baller or a teaspoon. Add the artichoke bottom to the bowl of water. Repeat with the remaining 3 artichokes.
Scatter the diced carrot in a large saucepan and arrange the artichoke bottoms on top. Scatter the potato and peas around the artichokes and nestle the onion in the center.
In a small bowl, whisk the olive oil with the flour, sugar and 1 tablespoon of the lemon juice. Stir in 1/4 cup of water and season with salt and pepper. Pour the mixture over the artichokes. Cover with a moistened piece of parchment paper and bring to a boil. Reduce the heat to moderately low and cook the vegetables until tender, about 1 hour. Check occasionally and add a few tablespoons of water as necessary if the vegetables begin to look dry.
Transfer the artichoke bottoms to a small platter. Stir the dill and the remaining 1/2 tablespoon of lemon juice into the vegetables in the pan and spoon them over the artichokes. Serve warm or at room temperature.