In a large dutch oven, heat 1 tbsp. EVOO over medium heat. Add the onion, partially cover and cook, stirring frequently, until golden-brown, about 15 minutes. Add 2 tbsp. EVOO, the apples, rosemary, 1/2 tsp. salt and 1/4 tsp. pepper; cook until the apples are crisp-tender, 5 to 7 minutes.
Meanwhile, in a large pot of boiling, salted water, cook teh linguine according to package directions; drain, reserving 1/2 cup of the cooking water.
Add the coleslaw mix and remaining 1 tbsp. EVOO to the dutch oven; increase the heat to medium-high and cook, stirring occasionally, until the slaw is lightly browned, about 5 minutes. Add the ham, linguine and reserved cooking water; cook, tossing, until heated through, about 3 minutes.