Pour the tomatoes into a large bowl and crush them with your hands.
In a large pot, warm ¼ cup/60 ml of the olive oil over low heat. Add the garlic and red pepper flakes and cook, stirring, until the garlic is fragrant and golden, about 2 minutes. Pour in the tomatoes (set the bowl aside), increase the heat to medium-low, and cook, stirring occasionally, until the tomatoes start to break down into a sauce, about 15 minutes.
While the tomatoes cook, break the stale bread slices into big chunks and place in the tomato bowl. Add cold water to cover and let stand until the bread has soaked up enough water to become soft again, about 10 minutes.
Remove the bread from the water, squeeze it between your hands to remove the excess water, and crumble into the tomato sauce. In a large measuring cup, stir together the tomato paste and warm water until well combined, then add to the pot. Season with the salt, reduce the heat to low, and cook for about 10 minutes, stirring vigorously from time to time with a whisk to give the pappa al pomodoro its typical creamy texture.
Remove from the heat, add the torn basil leaves and the remaining ¼ cup/60 ml olive oil, and stir to combine. Season to taste with additional salt as necessary.
Let the soup stand at room temperature for at least an hour to allow the flavors to mingle, then serve at room temperature or reheat gently over low heat to serve warm.