Preheat oven to 400°F and line a baking sheet with parchment paper. Rinse cod and pat dry. Line up fish on pan, season both sides with salt and pepper, and divide 1 tablespoon of butter evenly across pieces. Bake for 10 minutes. Flip cod and bake for another 8-10 minutes, until firm and flaky. Remove from oven and cool.
Add potatoes to a large pot with salt and bring to a boil over medium-high heat. Boil until tender and falling apart. Drain and rest until cool enough to handle.
Add cooled potatoes to a large mixing bowl and mash evenly. Crumble in cod fish in small chunks. Add green onion, parsley, thyme and mustard, along with remaining butter, melted. Add egg and stir to mix evenly, but without breaking the fish too small. Season to taste with salt and pepper. When ready, mixture should be sticky and hold together well.
Spread flour on a plate. Gather a large handful of the mixture and pack into a ball. Flatten to a disk and dust both sides with flour.
Heat oil in a nonstick pan over medium-high heat. Fry cakes until golden brown, about 5 minutes per side. Remove from oil and drain on a paper towel. Fish cakes taste great hot or at room temperature and can be made ahead and frozen. Reheat in the oven.
To make spicy mayo: mix together mayonnaise, hot sauce, mustard and parsley in a small bowl.