1. Preheat oven to 325°.
2. Heat a medium ovenproof saucepan over medium-high heat. Add 1 1/2 teaspoons oil; swirl to coat. Sprinkle pork with 1/2 teaspoon salt and 1/2 teaspoon pepper. Add pork to pan; cook for 8 minutes, browning on all sides. Discard oil from pan. Add 1/2 cup water, 1/2 cup stock, and soy sauce. Cover and bake at 325° for 2 hours or until very tender. Remove pork from pan; reserve liquid. Place pork on a cutting board; cover with foil. Let stand 10 minutes; cut into 4 slices.
3. While pork cooks, heat a Dutch oven over medium heat. Add 1 1/2 teaspoons oil; swirl to coat. Add sliced onion and 1/4 teaspoon salt; cook 2 minutes or until golden, stirring occasionally. Add 1/2 cup water, 1/2 cup stock, greens, and sugar. Bring to a boil. Cover; reduce heat, and simmer 25 minutes or until tender, stirring occasionally.
4. Remove 1 cup tomato liquid from can; reserve remaining liquid and tomatoes for another use. Combine 1 cup tomato liquid and flour in a medium bowl, stirring with a whisk. Place a zip-top plastic bag inside a 2-cup glass measure. Pour pork cooking liquid into bag; let stand for 5 minutes (fat will rise to the top). Seal bag; snip off 1 bottom corner of bag. Drain liquid into tomato mixture, stopping before fat layer reaches the opening; discard fat. Return the pan to medium-high heat. Add remaining 1 1/2 teaspoons oil to pan; swirl to coat. Add chopped onion and mushrooms; sauté for 2 minutes. Add tomato mixture to saucepan; bring to a boil. Cook until reduced to 1 cup (about 10 minutes).
5. While gravy cooks, bring remaining 1 cup water, 1 cup stock, 1/4 teaspoon salt, 1/4 teaspoon pepper, and garlic to a boil in a medium saucepan. Gradually add grits, stirring constantly with a whisk. Reduce heat; simmer 5 minutes or until liquid is absorbed, stirring frequently. Serve grits with gravy, pork, and greens.