For the cheesesteak filling: Saute the onions and peppers in the oil until translucent, and then remove from the pan. Add the mushrooms to the pan and cook until well browned. Return the peppers and onions to the pan along with the beef. Cook the beef, chopping up with a wooden spoon, until just cooked through. Season with salt, remove from the heat and spread out on a baking sheet to chill.
For the cheesesteak sauce: Combine the milk and cream and bring to a simmer. Whisk in the cheeses until thoroughly melted and well mixed. Set aside until ready to use.
For serving: For each serving, heat 8 ounces of the steak and vegetable mixture in a saute pan. Add 4 to 5 ounces of the sauce and bring to a simmer. Cook until heated through very well and the sauce has started to reduce. Once hot, pour into bowls and top with the rounds of baked puff pastry.