Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs. We introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with Hurst index s - 1/2. For a special choice of Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams. The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp. The global Gaussian fluctuations depend on s and coincide with the process in Kerov's CLT for s = -1/2. We give a dynamical explanation of this relationship using results of Eliashberg and Dubrovin. This is work in progress with Robert Chang (Rhodes College).