Heat olive oil in a large saucepan over medium-high heat. Add rice, onion, dill, and mint; sauté until onion is soft, about 5 minutes. Pour in 1/2 of the broth, reduce the heat to low, and simmer until rice is almost cooked through, 10 to 15 minutes. Stir in 1/2 of the lemon juice and remove from the heat.

Plunge grape leaves into a deep container of very hot water until soft, about 10 seconds; pat dry.

Place a grape leaf onto a work surface with the shiny side facing down. Place 1 teaspoon rice mixture on leaf at the stem end; fold both sides in toward the center and roll up from the wide bottom to the top. Place stuffed leaf into a 4-quart pot. Repeat to stuff remaining leaves, packing them into the pot tightly so they don't open while cooking.

Drizzle olive oil and remaining lemon juice over leaves, then pour in remaining broth to cover. Cover the pot and simmer for about 1 hour; do not let it boil as the stuffing may burst out of the leaves.