Place cottage ham in a large Dutch oven. Add cider, onions, thyme, garlic, bay leaves, cinnamon sticks, peppercorns, cloves, and enough water to cover cottage ham entirely. Cover, and place on high heat; bring to a boil, about 10 minutes. Once boiling, reduce heat, and cook on a low simmer for 1 hour and 30 minutes.
While ham is simmering, preheat the oven to 350 degrees F (175 degrees C). Cut cabbage into quarters, remove core, and cut into 1-inch slices. Place cabbage slices in a large bowl. Add olive oil, salt, and pepper; toss to coat.
Stir together honey mustard, stone-ground mustard, and brown sugar in a medium bowl. Once cottage ham has simmered, transfer ham to a roasting pan. Brush honey mustard mixture over entire cottage ham.
Bake in preheated oven until glaze has melted and a thermometer inserted in thickest portion registers 145 degrees F (63 degrees C), 20 to 30 minutes.
While ham is baking, reserve 1 1/2 cups of the cooking liquid in Dutch oven; discard solids and remaining liquid. Place Dutch oven over medium-high; add sliced cabbage, and stir to combine. Cover, and cook until cabbage is tender, about 15 minutes. Arrange cabbage on a platter.
Remove cottage ham from oven, and let rest for 10 minutes. Carve slices and serve on top of cabbage.