Place a colander in a 2-quart glass measure or medium bowl. Line colander with 4 layers of cheesecloth, allowing cheesecloth to extend over outside edges. Spoon yogurt into colander. Cover loosely with plastic wrap; refrigerate 12 hours or overnight. Spoon yogurt cheese into a bowl; discard liquid. Cover and refrigerate.
Combine gooseberries and sugar in a medium nonstick skillet. Cook over medium heat 5 minutes or until mixture begins to thicken, stirring frequently. Gently mash gooseberries with a fork; simmer an additional 2 minutes. Spoon into a small bowl; cover and chill.
Place whipping cream in a large bowl; beat with a mixer at high speed until stiff peaks form. Fold in yogurt and gooseberry mixture. Serve chilled.