Do Ahead: Dip can be made 3 days ahead. Cover and chill.
2 bunches green-stemmed Swiss chard (about 1½ lb.)
⅔ cup extra-virgin olive oil, divided, plus more
5 garlic cloves, finely chopped
½ cup tahini
⅓ cup fresh lemon juice
Toasted flatbread and lemon wedges (for serving)
Remove ribs and stems from Swiss chard leaves and finely chop. Tear leaves into small pieces. Set both aside separately.
Heat ⅓ cup oil in a large pot over medium-low. Cook reserved ribs and stems, stirring often and adding a splash of water if they start to brown, until tender, 5–7 minutes. Add garlic; cook, stirring, until fragrant, about 1 minute. Add reserved chard leaves by the handful, letting them wilt before adding more; cook, tossing, until all the leaves are wilted and tender, 10–12 minutes total. Let cool. Squeeze excess liquid from mixture into a measuring glass. (You should have about ½ cup liquid.)
Place Swiss chard mixture and 1 Tbsp. cooking liquid in a food processor and add tahini, lemon juice, and ⅓ cup oil. Season with salt and process, adding more cooking liquid if needed, until dip is creamy and only speckles of chard remain. This could take up to 5 minutes. Season with more salt if needed.
Transfer dip to a serving bowl and drizzle with more oil. Serve with flatbread and lemon wedges.
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