Place peppers and tomatoes in a medium-sized saucepan or pot. Pour in 1/2 cup water and turn heat to high. Season with 1 teaspoon kosher salt and pepper to taste. Bring to a simmer, then turn heat down to medium high. Set a timer for 25 minutes.
After about five minutes, the tomatoes and peppers will begin to release their juices, and the whole mixture should be bubbling. Adjust the heat if necessary so that the mixture stays at a constant bubble — medium to medium-high should do it. Stir every five minutes or so to make sure the tomatoes and peppers are not sticking to the bottom of the pan. If they are, add water by the 1/4 cup.
When the peppers and tomatoes are tender and nearly all of the liquid has evaporated and the tomatoes and peppers are beginning to stick to pot, add the basil, butter, and oil to the pot, give it a stir, then transfer the contents of the pot to a food processor or blender. Blend until smooth. Taste. Adjust seasoning with more salt and pepper as necessary.