Soak porcini in boiling water in a heatproof bowl, 20 minutes. Lift out, rinse off grit and chop finely. Strain and reserve soaking liquid through a damp paper coffee filter in a sieve.
Heat 2 tablespoons oil in a soup pot over medium-low. Add onions and 1/2 teaspoon salt. Cook 35–40 minutes, stirring often, until soft and deeply golden.
Meanwhile, heat remaining oil in a large skillet over medium-high. Brown fresh mushrooms in 2 batches, 5–6 minutes each. Leave briefly undisturbed between turns.
Stir garlic into onions for 30 seconds. Add sherry; simmer 1 minute. Add porcini, filtered soaking liquid, broth, thyme, browned mushrooms and remaining salt.
Simmer 10 minutes. Add kale and pepper; simmer 7–8 minutes until kale is tender. Stir in vinegar.