In a 5- to 6-quart slow cooker, stir together ½ cup ketchup, sugar, garlic, sage, oregano, 1 teaspoon coarse salt, and ½ teaspoon pepper.

Cut pork in half lengthwise; add to slow cooker, turning to coat.

Cover, and cook on low until meat is very tender and falling apart, about 8 hours (or on high for 6 hours).

Transfer pork to a large bowl. Using a large spoon, skim off and discard any fat from surface of cooking liquid. With two forks, pull meat apart until shredded.

Pour any juices from slow cooker over pork, add remaining ¼ cup ketchup, and stir to combine; season with salt and pepper.

Spoon meat onto bottom half of rolls; add red cabbage, if desired, and top of roll.

Preheat oven to 350°F. Stir together ½ cup ketchup, sugar, garlic, sage, oregano, 1 teaspoon coarse salt, and ½ teaspoon pepper in 5-quart Dutch oven or heavy pot with a tight-fitting lid.

Cut pork in half lengthwise; add to Dutch oven, turning to coat. Add 1 cup water, and cover.

Transfer to oven; cook until pork is tender, about 2 ½ hours.

Transfer pork to a large bowl. Using a large spoon, skim off and discard any fat from surface of cooking liquid. With two forks, pull meat apart until shredded.

Pour any juices from slow cooker over pork, add remaining ¼ cup ketchup, and stir to combine; season with salt and pepper.

Spoon meat onto bottom half of rolls; add red cabbage, if desired, and top of roll.

In a medium bowl, toss 1 ½ cups shredded red cabbage with 2 tablespoons cider vinegar and a little coarse salt. Let stand until soft and bright in color, at least 30 minutes and up to overnight.