In the bowl of a food processor or blender, purée red peppers and 2 cups broth until smooth. Transfer to a large saucepan and add remaining 2 cups broth, oregano and bay leaves. Stir to combine, heat on medium-high and bring to a gentle boil. Decrease heat to medium-low and simmer, stirring occasionally, for 2 minutes.
Meanwhile, coat a large grill pan or griddle with fat of choice (FOC) and heat on medium-high. Season shrimp with black pepper. Add shrimp to 1 side of pan and cook for 1 to 2 minutes per side, until opaque and cooked through. Transfer to a large plate.
Remove soup from heat. Remove and discard bay leaves; stir in basil. Ladle soup into serving bowls, top with shrimp. This recipe can be frozen & used later.