One of the simplest questions in riverine science remains unanswered: "What controls the width and depth of rivers?". This question has long been one of concern to scientists who seek to understand both the first-order and higher-order complexities of riverine morphodynamics, as well as to those who use morphological features to interpret past and present climates of Earth and other planetary bodies. Alluvial rivers, rivers that have beds and banks composed of mobile sediment, present an opportunity to understand the relationship between river discharge, bank material, and channel form. A wealth of theoretical and empirical work has demonstrated that coarse-grained rivers (average bed grain size > 10mm) self-organize their hydraulic geometries such that fluid shear stresses in the channel are slightly in excess of the threshold of motion for the bed sediment. By contrast, and in spite of their global prevalence, there exists no satisfactory theory to explain the controls on the hydraulic geometries of fine-grained river systems (average bed grain size < 1mm). To address this, we combine analysis of global channel geometry data sets in combination with examination of a longitudinal river profile as it transits from gravel to sand-bedded, to propose that alluvial rivers adjust their geometry to the threshold-limiting material: the structural component of the river channel that is the most difficult to erode. For coarse-grained rivers it is gravel, but for sand-bedded rivers it is mud (if present). Thus, for gravel-bedded rivers, given that the critical shear stress for bed material is typically larger than that of muddy/sandy banks, their first-order hydraulic geometry may be understood without considering bank composition. For sand-bedded rivers, however, we posit that cohesive bank material is crucial for setting hydraulic geometry. We have developed a novel instrument, designed explicitly for ease of implementation in the field, that is capable of determining the critical shear stress of cohesive sediments in-situ. By directly controlling the fluid shear stress exerted on the substrate, and detecting the onset of erosion by monitoring abrupt changes in the turbidity of the eroding fluid, we are able to determine the threshold of motion for a given cohesive substrate. We use this instrument to test our hypothesis that all alluvial river channels, regardless of grain size, adjust their cross-sectional geometry such that bankfull fluid shear stress is close to the critical shear stress associated with the threshold-limiting material. We have conducted a field investigation on a river in the New Jersey coastal plain to contrast direct measurements of bank toe erodibility against estimations of bankfull shear stress, back-calculated from surveys of channel geometry. From this, we demonstrate that the critical shear stress of the cohesive bank toe material is the first-order attractor for the hydraulic geometry of fine-grained alluvial rivers. We utilize this framework to cast the controls on braided versus single-threaded planform morphologies in terms of a channel's discharge, slope, and threshold-limiting material erodibility, and demonstrate that the presence of cohesive sediment in river banks allows for single-threaded planform morphologies in the overwhelming majority of alluvial river systems.