Slice the leek lengthwise then chop it up. Add the chopped leeks to a bowl of cold water and rinse in a few changes of water until all the grit is gone.
Place a medium size heavy bottom soup pot on medium flame. Add a lug of olive oil and sauté the cleaned leeks with a pinch of sea salt for about 40 minutes until completely wilted and they start to caramelize.
Add the vegetable stock to the pot and bring to a simmer. Add in the green peas and cook everything together for a few minutes until the peas are warmed through.
Using an immersion blender puree the leek soup to your liking. I go for a rustic hearty texture but you do as you please.
Season to your taste with the sea salt.
Meanwhile to make the chive oil, add the chives and 3 tablespoon of olive oil to a small food processor and puree until smooth. Season with a tiny pinch of sea salt.
Use half of the chive oil to stir in the soup and reserve the rest to drizzle over the top.
Laddle the leek soup into bowls and serve with a poached egg on top and a drizzle of the chive oil. Garnish with some edible flowers and a few flakes of coarse smoked sea salt.