Heat a large skillet over medium-high heat with 2 turns of the pan of EVOO. Add the andouille and cook until lightly browned, about 2-3 minutes. Remove the browned andouille to a mixing bowl and reserve rendered andouille fat in the skillet.
In a food processor add the beans, egg, flour, salt and pepper. Pulse until just combined -- you don't want it completely smooth.
In the mixing bowl with the andouille, combine the bean mixture with the rice, thyme, paprika, cayenne, cumin and the leftover rice. Form into 4 equal size patties. Heat the skillet with the andouille fat in it over medium-high heat. Once hot, add the patties and cook on each side for 7-8 minutes, until brown and heated through.