Prep the vegetables: slice leeks, dice potatoes, carrots, and celery, and mince the garlic.
Warm olive oil and butter in a large pot over medium heat.
Cook leeks with a pinch of salt until soft and sweet, about 6–8 minutes, stirring often.
Stir in celery and carrots; cook 3 minutes. Add garlic and cook 30 seconds until fragrant.
Add potatoes, ham, thyme, and black pepper. Toss to coat the vegetables and ham in the aromatics.
Pour in broth. Bring to a boil, then reduce heat to maintain a gentle simmer.
Simmer until potatoes are tender, 15–20 minutes. Skim any foam if needed.
Lightly mash some potatoes in the pot or blend briefly for a creamier texture.
Stir in cream and parsley; warm without boiling. Season with salt and lemon juice to taste and serve hot.