Chop the green onions and green garlic and separate the white and light green parts from the green tops. If you are using regular garlic cloves, put them with the whole parts of the green onions. Slice the potatoes and artichoke hearts into chunks you would want to eat, with a spoon. In a large pot set over medium-high heat, warm the olive oil for 1 minute. Add the white parts of the green onions as well as the garlic and stir-fry for 1 minute. Add the potatoes, stir to combine and cook 1 minute. Add the diced tomatoes with their liquid and the quart of vegetable stock.
Bring to a simmer, add salt to taste; then cover and cook over medium-low heat for 10 minutes. Add the artichoke hearts and cook another 5 minutes. Add the chickpeas and green peas and cook another 5 minutes. Remove the cover from the soup and add the asparagus. Cook 2 minutes. Add the greens and the green parts from the green onions and green garlic, if using. Stir well to combine and cook 1 minute. Turn off the heat and stir in the pesto. Serve topped with grated cheese.