Add barley to boiling salted water and cook until tender, about 30 to 45 minutes. Add the broccoli the last 3 minutes of cooking and cook until tender yet still slightly crisp. Drain both in a colander in the sink and rinse under cold water to stop the cooking. Drain well then add to a large mixing bowl.
Add olive oil to a medium skillet over medium high heat. Once hot, add the garlic and almonds and stir until the garlic and almonds are fragrant, about 1 minute. Turn off heat, sprinkle in the smoked paprika, season with salt and pepper, and stir for 30 more seconds. The residual heat will toast the paprika.
Carefully stir in the vinegar. Pour the dressing over the broccoli and barley and toss well all together. Finally add in the arugula and the cheese, taste for seasoning, and adjust as necessary.