1. Add half of cipollini to a pot of boiling water; simmer 1 minute, then transfer to a colander with a slotted spoon and immediately rinse with cold water. Repeat with remaining cipollini, then pearl onions (in 2 batches), then garlic.
2. With a small, sharp serrated knife, trim root ends and peel onions and garlic on a cutting board, slitting each lengthwise and holding your thumb against the pointy tip to keep its core from squirting out; then trim any fibrous tips. Put onions and garlic in a bowl and drizzle with 1 tbsp. oil, the thyme leaves, pepper, and salt. Mix to coat.
3. In a large frying pan, heat remaining 2 tbsp. oil over medium-high heat. Add onions and garlic, and sauté, stirring often, until most are partly browned. Add vinegar and broth; reduce heat to medium-low and simmer until onions are tender and some are falling apart, 10 to 20 minutes. Stir in green onions and butter.
4. Using a slotted spoon, transfer onions to a serving bowl; cover. Boil sauce over high heat until syrupy and reduced to about 3/4 cup, about 4 minutes. Stir gently into onion mixture; top with thyme sprigs.
Make ahead: Chill airtight up to 2 days (add green onions and thyme sprigs only at end).
Note: Nutritional analysis is per serving.