In a small pot, combine rice, broth, raisins, and ¼ teaspoon salt. Bring to a boil. Cover and reduce heat to medium-low. Simmer until rice is tender, about 17 minutes.
Meanwhile, thoroughly rinse Swiss chard. Cut off the thick ribs and stems from the leaves and thinly slice. You should have about 2 cups of sliced stems and ribs. Then, cut any extra large leaves in half lengthwise.
Bring a large pot of salted water to a boil. Place a large bowl of ice water nearby. Once boiling, use tongs to dip Swiss chard leaves into the water for 15 seconds to blanch, being sure not to overcook. Then, transfer to the bowl of ice water to cool. Once cooled, drain leaves and arrange on a paper towel–lined plate to dry.
Empty water from pot and wipe dry. Heat pot over medium-high with 1 tablespoon of olive oil and add the ground beef. Season with salt and pepper and cook the meat, breaking up with a spoon, until browned, about 8 minutes. Using a slotted spoon, transfer beef to a large bowl. Discard excess oil.
Return pot to medium-high heat with 3 tablespoons of olive oil. Once the oil is hot, add the onion and sliced Swiss chard ribs and stems. Season with salt and pepper. Cook, stirring occasionally, until softened, about 5 minutes. Add garlic and cook, stirring, until fragrant, 30 seconds more. Transfer to bowl with ground beef.
Add the cooked rice to the bowl with the ground beef and Swiss chard stems and stir in dill, lemon juice, and capers. Season to taste with salt and pepper.
Working with one Swiss chard leaf at a time, top with a few tablespoons of filling, depending on leaf size, and roll into a cigar shape, tucking the ends. Arrange on a serving dish with lemon wedges.