Place the smoked haddock in a large bowl, and cover with cold water. Soak fish 4 hours, changing water every 30 minutes. Alternatively, keep bowl under a thin stream of constantly running water. If using salt cod, soak the fish 12 hours.
Remove haddock from water; discard any bones. Transfer fish to saucepan. Add milk, garlic, thyme and potato. Set over medium-low heat; simmer until haddock is very soft, about 1 hour. Pour mixture through a sieve set over bowl; discard thyme and milk. Transfer solids to a large bowl.
Heat the oven to 400 degrees. Using a fork, gently mash the haddock, potato, and garlic together, adding enough cream to make a thick paste. Add the salt and pepper, and stir in the horseradish.
Place the bread slices on a baking sheet, and toast until golden. Brush toast with olive oil, and spread fish paste on top. Garnish servings with the fresh thyme.