Heat 1 teaspoon of the olive oil in a Dutch oven over medium-high. Add reserved cup of asparagus tips. Cook, stirring occasionally, 2 to 3 minutes. Use a slotted spoon to transfer to a plate.
Reduce heat on Dutch oven to medium and add the remaining 2 tablespoons of olive oil to Dutch oven. Add the thinly sliced leeks and cook, stirring occasionally, until softened, 4 minutes. This step is required to promote browning but be careful not to burn the leeks. Add potatoes and minced garlic cloves and cook, stirring constantly, 1 minute. Add all of the unsalted chicken stock at once. Bring to a boil over medium-high. Cook until potatoes are semi tender, 6 to 8 minutes. Add kosher salt and uncooked asparagus pieces and cook 5 to 8 minutes.
Remove the pot from heat. Stir in chopped fresh parsley. Puree using an immersion blender until smooth. Stir in heavy cream and orange juice and place back on very low heat to warm through.
Ladle soup into individual serving bowls. Drizzle servings with heavy cream and top with the cooked asparagus tips. Garnish additionally, if desired, with chopped cooked bacon, additional parsley, a dusting of Parmesan cheese, croutons, or basic wedges of lemon passed tableside. Sea salt and olive oil also work well to be minimal while keeping the dish bright.