In this paper we introduce and investigate the notion of two-Lipschitz left-hand quotient operator ideals, extending classical operator ideal theory to the setting of two-Lipschitz operators acting between metric and Banach spaces. Our approach relies on the linearization property ( \(\textrm{LP}\) ) of two-Lipschitz operator ideals relative to suitable linear operator ideals. We establish that such quotients naturally generate new two-Lipschitz operator ideals. Moreover, under the \(\textrm{LP}\) assumption, these quotients coincide with composition ideals of the form \((\mathcal {A}^{-1} \circ \mathcal {B}) \circ \textrm{BLip}_0\) . As applications, we characterize Grothendieck and Rosenthal two-Lipschitz operators as left-hand quotients, analyze their isometric properties, and explore their structure. We conclude with an open problem concerning quotients that do not satisfy the \(\textrm{LP}\) , using \((p, p_1, p_2)\) -summing two-Lipschitz operators as a motivating example.
