Complex workpieces, particularly aeronautical parts made of solid stock presenting a bottom wall and at least one stiffening rib transverse to the bottom wall, are machined using a device having a tool head furnished with a transmission component rotatable about a relative longitudinal axis, and a tool, typically a milling cutter, connected to the transmission component and capable of piercing the transverse rib. The longitudinal axis of the tool is disposed obliquely to the longitudinal axis of the transmission component so that the stiffening rib can be pierced at a point close to the bottom wall of the workpiece. The device also incorporates a feeler mechanism associated with the tool head and designed to engage the bottom wall during the movement of the tool toward the machining location, in such a way that the position of the tool head can be actively controlled and the tool deployed at a selected distance from the bottom wall.