Cut pea pods in half lengthwise, leaving some peas on each side of pods. Combine peas, arugula, 1/4 cup basil, and 1/4 cup mint in a large bowl. Add oil and 2 tablespoons lemon juice and toss to coat. Season salad with salt and more lemon juice, if desired.
Tear open balls of burrata (if using buffalo mozzarella, slice 1/2" thick) and arrange on a platter. Top with salad and more basil and mint.