Cooper quartets represent exotic fermion aggregates describing 
correlated matter at the basis of charge-$4e$ superconductivity and 
offer a platform for
studying four-body interactions, of interest for topologically protected 
quantum computing and strongly correlated matter. Focusing on 
solid-state systems,
we show how to quantum design Cooper quartets in a double-dot system 
coupled to ordinary superconducting leads through the introduction of an 
attractive
interdot interaction. A fundamentally novel, maximally correlated 
double-dot ground state, in the form of a superposition of vacuum 
$|0\rangle$ and four-electron
state $|4e\rangle$, emerges as a narrow resonance in a many-body quartet 
correlator that is accompanied by negligible pair correlations. The 
results open
the way to the exploration of interaction effects in hybrid 
superconducting devices and novel correlated states of matter in a 
quantum solid-state lab.