In a medium saucepan, combine onions and water to cover; bring to a boil over medium-high heat until just tender, about 4 minutes. Drain and peel.
In a large skillet, melt butter and 2 tablespoons oil over medium-high heat. Add onions; cook until browned, about 4 minutes. Add garlic; cook, stirring occasionally, until fragrant, about 2 minutes. Season with salt and pepper. Remove from skillet and wipe it clean.
In same skillet, heat remaining 2 tablespoons oil over medium-high heat until oil begins to shimmer. Add mushrooms; cook, stirring constantly, until mushrooms release their liquid and become tender and brown, about 4 minutes. Transfer to plate.
In the same skillet, bring broth and cream to a boil over medium-high heat, stirring constantly, until mixture begins to thicken, about 1 to 2 minutes. Stir in onions and mushrooms; reduce heat to low, and simmer, stirring gently, until heated through, about 4 minutes. Stir in thyme. Serve immediately. Garnish with thyme, if desired.
Recipe by Southern Lady Magazine at https://southernladymagazine.com/creamed-pearl-onions-mushrooms/