One of the most important problems in the field of distributed optimization is the problem of minimizing a sum of local convex objective functions over a networked system. Most of the existing work in this area focuses on developing distributed algorithms in a synchronous setting under the presence of a central clock, where the agents need to wait for the slowest one to finish the update, before proceeding to the next iterate. Asynchronous distributed algorithms remove the need for a central coordinator, reduce the synchronization wait, and allow some agents to compute faster and execute more iterations. In the asynchronous setting, the only known algorithms for solving this problem could achieve an either linear or sublinear rate of convergence. In this paper, we build upon the existing literature to develop and analyze an asynchronous Newton-based method to solve a penalized version of the problem. We show that this algorithm guarantees almost sure convergence with a global linear and local quadratic rate in expectation. Numerical studies confirm the superior performance of our algorithm against other asynchronous methods.