In a large pot, saute the onion over medium high heat in the extra virgin olive oil until it becomes transparent and starts to become soft. Add the stock.
Then add the peas and potatoes, and stir well.
Bring to a boil, add salt; then reduce heat and simmer over low heat for 35 to 40 minutes or until peas are soft, stirring occasionally.
Remove from heat and blend until smooth, with immersion blender or a counter-top blender. Taste for salt and pepper; add as needed.
Add a splash of milk, cream, or half and half, if desired. (I frequently use half and half.) Serve split pea soup hot with extra pepper and crusty bread, or croutons.
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