Place potatoes in heavy large pot. Cover with cold water. Boil over medium-high heat until potatoes are tender when pierced with fork, about 40 minutes. Drain, reserving 1 1/2 cups potato cooking liquid. Return potatoes to dry pot. Stir over medium heat until any excess liquid evaporates. Add 4 tablespoons olive oil and mash until almost smooth. Mix in enough potato cooking liquid as needed to moisten. Stir in caramelized onions. Season to taste with salt and pepper. (Can be made 2 hours ahead. Let stand at room temperature. Rewarm over medium heat, adding more reserved cooking water as needed to moisten.).
To caramelize onions: Add 3 tablespoons olive oil to a saucepan and add onions. Saute over medium-high heat until onions are starting to caramelize 10-12 minutes. Add the garlic for the last minute of cooking. If you like your onions darker, saute them slightly longer (or if you like them lighter, diminish the saute time).
Note: A couple of tablespoonfuls of parmesan stirred into these potatoes is a lovely thing!