  We prove $L^p$ bounds for partial polynomial Carleson operators along
monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial
consisting of a single monomial. These operators are "partial" in the sense
that we consider linearizing stopping-time functions that depend on only one of
the two ambient variables. A motivation for studying these partial operators is
the curious feature that, despite their apparent limitations, for certain
combinations of curve and phase, $L^2$ bounds for partial operators along
curves imply the full strength of the $L^2$ bound for a one-dimensional
Carleson operator, and for a quadratic Carleson operator. Our methods, which
can at present only treat certain combinations of curves and phases, in some
cases adapt a $TT^*$ method to treat phases involving fractional monomials, and
in other cases use a known vector-valued variant of the Carleson-Hunt theorem.
