For the pork hash burger:
For the hot basil aioli:
Preheat grill or steamer.
Combine all of the pork hash burger ingredients in the bowl of a food processor and pulse until well blended and emulsified. Form into 6 even balls and flatten into patties. Grill or steam until internal temperature reaches 160°F. Add cheese to melt. Grill or steam burger buns if desired.
Meanwhile, combine all of the ingredients for the hot basil aioli in a bowl or food processor and mix until blended. Taste and adjust seasoning as needed.
To serve, smear a tablespoon of hot basil aioli on the bottom and top bun, then top with the pork hash cheeseburger, butter lettuce, tomato, and shaved white onion. Serve immediately.