Thoroughly wash and coarsely chop the mixed greens, discarding any tough stems. In a large kettle, combine the greens with 1 cup of cold water, bring to a boil, and cook until wilted, about 5 to 15 minutes. Drain, reserving the cooking liquid.

In a heavy pot, heat the bacon fat or corn oil over medium heat. Whisk in the flour and cook, stirring constantly, until the roux reaches a peanut butter color, 15 to 20 minutes. Add the onion, garlic, and green pepper, and cook until softened, about 5 to 10 minutes.

Stir in the reserved greens liquid and enough water to make 1-1/2 quarts total. Add the bay leaf, thyme, salt, black pepper, white pepper, and cayenne pepper. Bring to a boil.

Return the wilted greens to the pot along with the smoked ham and sausage. Bring to a gentle boil, then reduce heat to low and simmer for 1-1/2 to 2 hours, stirring occasionally. Remove the bay leaf before serving. Adjust seasonings to taste and serve over white rice.