To avoid discoloration, toss the sunchokes with a squeeze of lemon juice in a medium bowl. Reserve a quarter of the green apple for garnish and rub its sides the lemon to also avoid discoloration. With the remaining apple, peel it, cut out its core and chop it up loosely. Add it to the cubes of apple to the bowl with the sunchokes; toss so the apple cubes are covered in lemon, too.
In a medium saucepan, melt the butter over medium-low heat. Add the thinly sliced leek and cook until soft, about 5 minutes. Throw in the minced garlic and thyme and cook until fragrant, an additional minute or so.
Next, add the sunchokes and apple. Top the sunchokes and apple with a few pinches of salt and pepper. Pour in the chicken broth and bring the mixture to a simmer. Turn the heat down to low, and simmer, while the pot is covered, until the sunchokes and apple have softened, about 30-40 minutes.
While the soup is cooking, let's make the Italian parsley oil. Transfer the Italian parsley to the blender or food processor; pulse until chopped. Pour in the oil and blend until very smooth. Transfer the mixture to a small bowl and allow it to sit while the soup is cooking.
If you have an immersion blender, use it! If you don't (I don't), transfer the soup, in batches, and blend until very smooth. Transfer back to the pot, warm the soup over low heat and salt to taste.
Just before serving, strain the parsley oil, discarding the parsley pulp and reserving the oil. Cut the reserved quarter of the green apple into matchsticks. Divide the soup between bowls; garnish each soup a small handful of green apple and a drizzle of Italian parsley oil.