The sweet-tart taste of peaches provides a welcome if unexpected accompaniment to savory dishes like this one.
Peel and cut potato into 1-inch dice. Place in a small saucepan; add enough cold water to cover by 2 inches. Bring to a boil; reduce heat to medium-low, and simmer until potato is tender when pierced with a sharp knife, about 12 minutes. Drain; while potato is still hot, pass it through a ricer or food mill into a large bowl. Set aside.
Combine milk and thyme in a small saucepan, and bring to a boil over medium-high heat. Add fish fillets; cover pan, and reduce heat to low. Cook until fish is opaque and cooked through, about 12 minutes. Drain fish in a colander, reserving cooking liquid. Discard thyme.
Add fish to potatoes, and mash with a fork until fish is broken up into very small pieces and mixture is well combined. If mixture is too dry, add some of the reserved cooking liquid a tablespoon at a time. Season with salt and pepper.
Thinly slice one of the peaches into 18 slices, and set aside. Chop remaining peaches into 1/4-inch pieces. Place chopped peaches in a small saute pan, and add 1/4 cup water; season with salt and pepper. Bring to a boil over medium-high heat. Reduce heat to medium-low; cover, and simmer until peaches are soft, 3 to 5 minutes. Remove from heat; let cool slightly. Transfer to a blender or food processor, and puree until mixture is smooth.
On a clean work surface, divide fish mixture into six parts. Shape each part into a 1-inch-thick fish shape using your hands. Heat the oil in a medium saute pan over medium heat. Working in batches so as not to overcrowd the pan, add fish cakes, and saute until they are golden on both sides, 2 to 3 minutes per side. Using a slotted spatula, transfer to a large plate. Repeat with remaining fish cakes.
To serve, pour 1/4 cup peach puree on each plate, and use reserved peach slices (3 per plate) to make waves. Place a fish cake on each plate, and serve.
Any mild white fish, such as flounder, sole, or halibut, can be used in place of cod.