How to achieve a flexible consistency on ice cream? (photos attached)
Ice cream is smooth because churning breaks up ice crystals, and sugar, fat and egg yolks prevent the ice crystals from re-forming. Pure ice cream can be relatively elastic when frozen just by having lots of sugar, fat, and egg yolks in it. However, you cannot achieve this sort of elasticity in ice cream when it's above freezing without stabilizers like Xanthan Gum and Guar Gum. These are more effective emulsifiers/stabilizers than eggs and work at a higher temperature than eggs do. 

Many stabilizers are natural products, Xanthan gum is the product of bacterial formation and Guar gum is processed bean paste, so I personally have no problem with them and use them in my ice cream sometimes to improve the texture. Mostly I just use more egg yolks - my ice cream rarely last long enough to melt anyway. 

I think in this case the creaminess you describe is more of a mouth feel thing rather than a flavor, if you want to replicate it I'd suggest getting more eggs in your recipe and consider adding some guar gum.