Slice the Leek lengthwise in half, remove a couple of layers of the outer leaves so that you are left with the tender center leaves.
Cut off the root ends and the dark green tops and thinly slice the leeks into 1/4 inch slices so you are left with the white and light green portions of the leeks.
Place the sliced leeks in a bowl, fill with cold water and swish the leeks around to remove any dirt from between the leaves, pour the leeks into a strainer and drain well.
In a medium pot, melt 3 tablespoon of Butter over low heat and add the sliced leeks to the melted butter. Stir to coat the leeks in butter and simmer over low heat for about 20 minutes, stirring often until leeks are soft and tender without browning.
Once leeks are tender, add in the Green Peas, turn heat up to medium low and gently stir to incorporate the peas into the buttery leeks.
Cook the peas with the leeks for about 7-10 minutes, stirring often until cooked through. Test for seasoning at this stage and add Salt as needed.
To serve, place buttered peas and leeks in a serving bowl and top with remaining 1 tablespoon of Butter and Fresh Mint.