Bring 1 cup water to a boil in a pot. In a small bowl, submerge dried porcinis in boiling water and set aside to soak for about 5 minutes. Meanwhile:
Peel shallot and garlic and thinly slice.
Rinse chard, and strip off leaves. Roughly chop stems and leaves and keep separated.
Rinse green beans, trim ends and slice into 1-inch pieces.
Rinse potatoes and cut into ¼-inch cubes.

In the same pot, heat 1 tablespoon oil until shimmering. Add shallots and a pinch of salt and sauté until soft and translucent, 2 to 3 minutes. Stir in garlic and chard stems and cook until fragrant, about 30 seconds. Toss in green beans and potatoes and cook until they begin to soften, 3 to 5 minutes. Meanwhile:
Rinse zucchini, halve and chop into bite-sized pieces.
Drain and rinse cannellini beans.
Remove porcinis from soaking liquid — reserving liquid for later — and roughly chop.

Stir zucchini, cannellini beans, tomatoes, porcinis and reserved porcini soaking liquid into the pot, being careful to not include the grit that has fallen to the bottom of the bowl. Add 2 cups water, bring to a boil and reduce to a simmer. Cook until all veggies are tender, 14 to 16 minutes.

Pinch basil leaves off stems, roll together, and cut into strips. Stir basil, chard leaves and pecorino into pot and cook until chard is wilted and cheese is melted, 1 to 2 minutes. Add ½ teaspoon vinegar, if you have it. Season with salt and pepper to taste. Divide evenly between two bowls and serve.