Using a novel instrument previously reported by our group [1], we have been able to perform Local Anodic Oxidation (LAO) over monolayer graphene to create unprecedented large areas (larger than mm2) of single layer Graphene Oxide (GO) with any shape and in a very short time (a few minutes). Our instrument is equipped with three electromechanical steppers with 0.05 µm accuracy and 1 µm precision, and each stepper with a linear piezoactuator. It operates at speeds up to 15 mm/s, and includes a steel/Ni spring probe of typically 6 µm radius to work in tapping or contact mode. The equipment is provided with a humidity control system that allows maintaining a stable Relative Humidity (RH) environment, together with a system of supply and electrical monitoring. Thus, electrical signal coming from the tip-sample interaction is continuously monitored (by means of a high speed oscilloscope). When the tip approaches to graphene on a wet atmosphere (RH higher than 40%), it is commonly accepted that a water bridge is created between them. When a negative bias is applied to the tip respect to the grounded graphene (in our case we operate between 20V - 40 V), the LAO phenomena takes place [2] (Fig. 1a). The process firstly consists on electrical water ionization in H+ and OH- ions. Hydroxyl anions are then conducted to the graphene surface, which is thus oxidized. Within a specific RH range, the oxide patterns created show a good reproducibility and regular shape (Fig. 1b). Controlling the movement of the tip, we can draw the desired patterns over the graphene or any other 2D-substrate (Fig. 1c). We have extensively studied and optimized the influence of the experimental variables affecting the process, like applied bias, contact time or relative humidity. This LAO procedure is so a very powerful tool for the fabrication of electrical devices. Also, the possibility of chemical functionalization of GO allow us to design a large variety of utilities like (bio)chemical sensors, supported catalysts, etc. In addition, we have been able to record the electrochemical trace generated during the oxidation reaction of graphene. To our knowledge, it is the first time these records have been reported. Moreover, based on these data, we have formulated a theoretical model based just on classical first principles that predicts the behavior of the LAO process over graphene. This model shows a very accurate fit for the dependence of the oxide pattern expansion with the applied voltage (Fig. 2a), with the contact time (Fig. 2b), and also for the evolution of the recorded transient electric current (Fig. 2c). This model reveals very relevant information about the processes governing LAO. First, at the beginning (< 20 ms), most of the total current I(t) passing through the tip-substrate junction does not come from the electrochemical reaction but from residual conductance of the non-oxidized graphene. We call this component drift current, IT(t). Second, for longer times (>200 ms), the total current becomes dominated just by the electrochemical reaction, and the drift component almost vanishes. We call this component electrochemical current, IEC(t) (Fig. 2c). Third, the process is mainly driven by the electric field increase at the edge of the pattern. And fourth, the GO generated has a very high oxidation degree. This work opens new promising fields for development of extensive fabrication methods, technical applications and the study of anodic oxidation processes of conducting and any 2d material at large scales.