Place red peppers in a blender or food processor and process until smooth. Set aside.
Heat oil in a large saucepan over medium-high heat. Add onion and garlic and cook, stirring, 1 minute. Add mushrooms and cook until softened and until liquid is released, about 3 minutes. Stir in thyme, oregano, and bay leaves.
Add red pepper puree and broth and bring to a boil. Reduce heat to medium, partially cover pan and simmer for 5 minutes. Add milk and simmer 1 minute to heat through.
Remove from heat, remove bay leaves and stir in basil. Season to taste with sale and cayenne. Ladle soup into bowls and top with feta. Serve warm or chilled.