Heat the olive oil in a large saucepan over medium low heat. Cook the onions and garlic, stirring frequently, for about 4 minutes, or until they begin to soften.
Add the bell peppers, cover, and cook, stirring occasionally for 10 to 15 minutes, until softened. Use a slotted spoon to remove the onions and bell peppers to a separate dish, reserve the olive oil.
Place the walnuts, spices, and lemon juice in a food processor or blender and process until the mixture is smooth.
With the motor running, gradually pour in the reserved olive oil. Add the onion and bell pepper mixture, process until the mixture reaches your desired consistency.
Taste and adjust the lemon juice, olive oil, salt, and pepper as desired.
To serve, transfer to a bowl and garnish with chopped walnuts, olive oil, parsley, and paprika if desired.
Store the Muhammara in an airtight glass container in the refrigerator for up to one week.