Drain the crab in a fine mesh strainer and check for any bits of shell to remove. While it's in the strainer, press the crab with your fingers or a potato masher to release excess liquid. Transfer the crab to a small mixing bowl.
If using rolled oats instead of quick oats, measure out ½ cup and pulse them in a food processor or blender to make them more fine. Gently stir the oats into the crab.
Peel a shallot and finely chop about ⅔ of it, until you have ¼ cup chopped shallot. Melt the butter in a large skillet over medium-high heat and cook the shallots in it until they are soft. Add the buttery shallots to your crab mixture.
Beat an egg in a tiny bowl or measuring cup and add it to the bowl with the crab. Add the pepper and gently stir to combine.
With clean hands, shape the crab mixture into 6 patties about 2 ½ inches wide. Add enough oil to your skillet to cover the bottom of the pan. Heat it on medium high until hot and then carefully add the crab cakes to the pan in a single layer, without any touching the others. (You might need to do this in two batches, if your pan isn't large enough to fit them all comfortably. If so, add more oil for the second batch if there isn't enough oil to cover the bottom of the pan.)
Cook the crab cakes for three minutes on the first side, and gently nudge them to see if they slide over easily. If so, the bottom is cooked and crispy enough to flip them over. Cook the second side for about three minutes, until the bottom is crispy. Drain the crab cakes on a plate lined with paper towels.
Serve the crab cakes with lemon wedges, and mayo, if desired. Garnish with parsley (optional).
Store leftovers in the refrigerator for up to three days.