Preheat the oven to 425°F.
Combine the onions, shallots, garlic and thyme with the olive oil and water on a large sheet pan that is at least 1-2" deep. Season with salt and pepper. Toss to coat.
Mound the ingredients in the center of the pan rather than spreading out; this will allow the, to steam without browning too fast.
Bake, stirring and tossing the onions every 10 minutes, until the onions are golden brown and starting to soften, about 30 minutes.
Discard the thyme sprigs and add the wine to the pan; stir and scrape up any browned bits from the bottom of the pan. Spread the onions out across the sheet pan in a thin layer.
Reduce the oven temperature to 400°F and return the mixture to the oven. Continue to roast for another 10-25 minutes longer, depending on how thick the onions were sliced, until they are completely caramelized and deeply golden brown. Some onions will be more deeply caramelized than others, this is ok.
Transfer the onions and garlic to a cutting board. Squeeze the roasted garlic from its peel and mash into a paste with the flat side of a chef’s knife or with a fork then transfer to a medium-sized bowl.
Finely chop the caramelized onions and add to the bowl with the garlic.
Stir in the yogurt, sour cream, and lemon juice. Season with salt and lots of black pepper. Transfer to a serving bowl and sprinkle the top with paprika, chives, and Everything Bagel Seasoning, if using, just before serving.