Heat butter and olive oil in a large 12-inch skillet over medium heat.
Add prepared Brussel sprouts to the skillet and cook for 4 minutes over medium-high heat, stirring occasionally.
Stir in sliced apples, salt, pepper, and rosemary; cook for 4 to 5 minutes, or until tender, stirring frequently.
In the meantime, prepare the walnuts by starting to melt the butter in a saucepan.
Add sugars to the melted butter and stir until incorporated; mix in walnuts and continue to stir around for about a minute.
Remove Brussels sprouts and apples from heat and let cool for a minute.
Spoon salad onto serving plate and garnish with candied walnuts and pomegranate arils.