For the pimento cheese: Combine the cream cheese and mayonnaise in the bowl of a stand mixer fitted with a paddle attachment. Mix on medium speed until smooth, about 1 minute. Add the Cheddar and Gouda and mix to combine on low, about 1 minute. Stir in the roasted red peppers by hand. Taste and season with salt and pepper, but go heavy on the pepper; start with 1/2 teaspoon and go from there.
For the sandwiches: On low heat, warm a cast-iron skillet. Put a heaping 1/2 cup of the pimento cheese between 2 slices of bread. Don't spread the mixture all the way to the crust because it will soften and spread as it heats in the skillet. Repeat with the remaining bread and cheese.
Smear one side of each sandwich with mayonnaise, and place the sandwiches in the skillet mayonnaise-side down. Smear the top pieces of bread with mayonnaise. Cook until golden brown and the cheese has just started to melt, about 5 minutes. Flip and cook the other side until golden, another 3 minutes. Slice into halves or quarters and serve immediately.
Reserve any remaining pimento cheese in the refrigerator for up to 3 days.