Set a large deep skillet with a lid over medium-high heat.

Brown the beef: Add the olive oil to the skillet and let it warm for a few seconds.

Add the ground beef to the skillet. Cook for 5 to 6 minutes, breaking it into small pieces with a wooden spoon, until no pink remains and the edges start to brown.

Use a slotted spoon to transfer the beef to a plate. Leave about 1 tablespoon of fat in the skillet.

Cook the vegetables: Add the chopped onion, diced carrot, and sliced mushrooms to the skillet. Cook over medium heat for 6 to 8 minutes, stirring often, until the onion softens and the mushrooms lose their moisture and start to brown.

Add the garlic, tomato paste, and smoked paprika to the skillet. Stir for 1 minute, until the garlic smells fragrant and the tomato paste looks a shade darker.

Add the rinsed pearl barley to the skillet. Stir for about 30 seconds so the barley is coated in the fat and seasonings.

Return the browned beef and any juices from the plate to the skillet.

Pour the beef broth into the skillet. Add the 1 teaspoon kosher salt and 1/2 teaspoon black pepper.

Raise the heat to high and bring the liquid to a boil.

Simmer the barley: Cover the skillet, lower the heat to a gentle simmer, and cook for 28 to 32 minutes. Stir once halfway through so the barley does not stick to the bottom.

Uncover the skillet and check the barley. It should be plump and tender with a little thick liquid left in the pan; if it still feels firm, add 1/4 cup water, cover again, and cook for 5 more minutes.