1. Place the cod in a large container and cover with cold water. Refrigerate for 24 hours, changing the water every 2 to 4 hours. Remove the cod from the water, squeeze out the excess liquid and roughly chop the bacalao.

2. In a medium saucepan set over high heat, add the potato and cover with water. Bring to a boil and cook until soft (a fork should easily pass through the potato without resistance), about 8 to 10 minutes. Turn the potato out into a colander to drain, then set the colander over a medium bowl. 

3. Rinse and dry the saucepan and set over medium-low heat. Add the olive oil and garlic and cook until the garlic is golden brown, stirring often, 5 to 8 minutes. Add the bacalao and cook about 15 minutes. Turn the mixture out into the colander with the potato and drain (the bowl will catch the oil); reserve the oil.

4. Transfer the bacalao mixture to the bowl of a food processor and pulse to roughly chop; purée, slowly adding the reserved garlic oil until the brandada is soft and spreadable but not soupy or runny. Season with salt as needed and transfer to a small oven-safe baking dish or four 6-ounce ramekins. 

5. Preheat the broiler to high. Place the brandada in the oven and broil until golden brown, 3 to 5 minutes (broiler intensities vary, so watch the broiler very carefully). Serve immediately with crackers.