Heat olive oil in a large pot over medium heat. Add the carrot, onion, and celery. Cook for 10–15 minutes, stirring occasionally, until softened and lightly caramelized. The onions should take on some color—this will give the soup lots of flavor and add a bit of sweetness.
Add the rinsed lentils and stir to coat in the oil. Let them toast slightly for about 3 minutes. Stir in the dried thyme.
Add the bay leaves, chicken bouillon paste, water (or stock), and diced potato. Bring to a simmer, cover, and cook for 20–30 minutes, until the lentils are very soft and the potatoes easily fall apart when pressed.
Remove the bay leaves. Leave the soup chunky or blend until smooth, depending on your preference.
While the soup cooks, combine parsley, olive oil, lemon zest and juice, salt, and pepper in a bowl. Stir well to combine.
Ladle the soup into bowls and top with parsley oil and crumbled feta. Serve with toasted bread rubbed with garlic and drizzled with olive oil.