  In the large top-mass limit, Higgs plus multi-gluon amplitudes in QCD can be
computed using an effective field theory. This approach turns the computation
of such amplitudes into that of form factors of operators of increasing
classical dimension. In this paper we focus on the first finite top-mass
correction, arising from the operator ${\rm Tr}(F^3)$, up to two loops and
three gluons. Setting up the calculation in the maximally supersymmetric theory
requires identification of an appropriate supersymmetric completion of ${\rm
Tr}(F^3)$, which we recognise as a descendant of the Konishi operator. We
provide detailed computations for both this operator and the component operator
${\rm Tr}(F^3)$, preparing the ground for the calculation in $\mathcal{N}<4$,
to be detailed in a companion paper. Our results for both operators are
expressed in terms of a few universal functions of transcendental degree four
and below, some of which have appeared in other contexts, hinting at
universality of such quantities. An important feature of the result is a
delicate cancellation of unphysical poles appearing in soft/collinear limits of
the remainders which links terms of different transcendentality. Our
calculation provides another example of the principle of maximal
transcendentality for observables with non-trivial kinematic dependence.
