As much as I love that movie its completely wrong.
White bread is the only choice for grilled cheese. Sourdough is amazing but is wrong here. His cheese mixture is pretty lame too. Gruyere is perfect and you only need one cheddar. Parmesan is too sharp and salty and will overpower the other cheese. A better mix is Gruyere for texture, cheddar for some mild flavor and then the third cheese is more to taste and what you want to be the forward taste. Things like provolone, colby, muenster, monterey jack, goudar, etc.
if you want to boost your grilled cheese, put some cheese in the pan and let it melt then put the sammich in that and let it grill on the sides. Then youll get the perfect mix of gooey cheese on the inside and grilled cheese outside.