Heat a large pot over medium-high. Add 1 to 2 tablespoons of olive oil along with the onion and cook, stirring, until softened, 2 to 3 minutes. Add the garlic, celery, half the dill and ½ teaspoon salt and stir until fragrant, 1 to 2 minutes.
Add the pearl barley and stock and bring to the boil. Cover, reduce heat to medium and cook, stirring occasionally, until the barley is swollen and tender, 25 to 30 minutes.
Turn off the heat. Add the lemon juice, and season with salt and pepper. Taste and make sure the seasonings are to your liking, adjusting salt and lemon as needed.
Stir in the spinach, allowing the residual heat to wilt the leaves.
Ladle the soup into bowls. Top with a dollop of yogurt and the remaining dill and drizzle generously with olive oil.
