Garlic Soup With Potatoes and Poached Eggs

Name: Garlic Soup With Potatoes and Poached Eggs
Ingredients: 2 tablespoons olive oil, plus more for serving; 4 tablespoons salted butter, divided; 3 heads of garlic, cloves peeled; 1 large leek, halved lengthwise, very thinly sliced, dark green parts reserved separately; 1 tablespoon thyme leaves; 1 1/2 pounds Yukon Gold potatoes (about 2 large), peeled, cut into 1-inch pieces; 4 cups (or more) low-sodium chicken broth; 2 tablespoons crème fraîche or sour cream, room temperature; Kosher salt; Freshly ground white or black pepper; 4 large eggs, room temperature; 2 tablespoons parsley leaves with tender stems; Finely grated lemon zest (for serving)