Use a potato masher or fork to mash the mixture until it is mostly broken down but still has some texture. Do not over-mash into a paste.
Add the egg (or flax egg) to the mixture and mix well until everything is just combined. The mixture should be firm enough to shape.
Divide the mixture into four equal portions. Form each portion into a patty about 3/4 inch thick.
Place the formed patties on a plate lined with parchment paper and chill in the refrigerator for at least 30 minutes. This helps them hold their shape.
Cook the patties in a lightly oiled skillet over medium heat for 5 to 7 minutes per side, until heated through and browned. Alternatively, grill over medium heat for 6 minutes per side.
Notes
For a gluten-free option, use certified gluten-free rolled oats.
If you prefer not to use liquid smoke, increase the smoked paprika to 1 1/2 teaspoons.
These patties freeze well after chilling; wrap individually before freezing.