Chop the almonds finely in a blender or food processor, then add 3 cups milk and blend until the mixture is smooth and creamy. Put this into a large pot and add the remaining 6 cups milk, sugar and vanilla. Bring to boil, lower the heat and simmer for 15 minutes, stirring constantly to prevent sticking.
Dissolve the cornstarch in 1-1/2 cups of cold water. Add this to the pot in a steady stream, stirring constantly. Continue to simmer and stir for another 15 minutes.
Transfer the pudding into a large serving bowl, or into individual dishes. Chill completely before serving.