Obtaining precise estimates of quantum observables is a crucial step of variational quantum algorithms. We consider the problem of estimating expectation values of Hamiltonians, obtained on states prepared on a quantum computer. Locally-Biased Classical Shadows is a novel estimator for this task, which is locally optimized with knowledge of the Hamiltonian and a classical approximation to the underlying quantum state. It is based on the concept of classical shadows of a quantum state, and has the important property of not adding to the circuit depth for the state preparation. The performance has been tested numerically for molecular Hamiltonians of increasing size, finding a sizable reduction in variance with respect to current measurement protocols that do not increase circuit depths. It has also shown its versatility when techniques to reduce circuit sizes of variational quantum eigensolvers are also implemented.