Step 1

Heat 2 Tbsp. extra-virgin olive oil in a large pot over medium-high. Arrange 2 small onions, unpeeled, quartered, and 1 head of garlic, unpeeled, halved crosswise, cut side down, in pot; add two 1" pieces ginger, scrubbed, smashed. Cook, turning occasionally, until cut sides of onions and garlic are deep brown, 5–10 minutes. Add 2 Tbsp. double-concentrated tomato paste and cook, stirring occasionally, until aromatics are coated and tomato paste is dark red, about 3 minutes.

Step 2

Add 3 celery stalks, very coarsely chopped, 1 medium carrot, scrubbed, very coarsely chopped, one 4x3" piece dried kombu, a few sprigs parsley, 3 oz. dried shiitake mushrooms, 1 tsp. black peppercorns, and 3 quarts water. Bring to a boil, then reduce heat and simmer, skimming foam from surface as needed, until stock is darkened in color and reduced by two thirds, about 2½ hours. Strain through a fine-mesh sieve into a large heatproof bowl, pressing on solids to extract as much liquid as possible; discard solids.Do Ahead: Stock can be made 1 week ahead. Cover and chill, or freeze up to 8 months.