Remove ribs and stems from 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; continue to stir for 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.