  An important open puzzle in the superconductivity of UTe$_2$ is the emergence
of time-reversal broken superconductivity from a non-magnetic normal state.
Breaking time-reversal symmetry in a single second-order superconducting
transition requires the existence of two degenerate superconducting order
parameters, which is not natural for orthorhombic UTe$_2$. Moreover,
experiments under pressure (Braithwaite et. al., Comm. Phys. \bf{2}, 147
(2019), arXiv:1909.06074 [cond-mat.str-el]) suggest that superconductivity sets
in at a single transition temperature in a finite parameter window, in contrast
to the splitting between the symmetry breaking temperatures expected for
accidental degenerate orders. Motivated by these observations, we propose a
mechanism for the emergence of time-reversal breaking superconductivity without
accidental or symmetry-enforced order parameter degeneracies in systems close
to a magnetic phase transition. We demonstrate using Landau theory that a cubic
coupling between incipient magnetic order and magnetic moments of Cooper pairs
(pair-Kondo coupling) can drive time-reversal symmetry breaking
superconductivity that onsets in a single, weakly first order transition over
an extended region of the phase diagram. We discuss the experimental signatures
of such transition in thermodynamic and resonant ultrasound measurements. A
microscopic origin of pair-Kondo coupling is identified as screening of
magnetic moments by chiral Cooper pairs, built out of two non-degenerate order
parameters - an extension of Kondo screening to unconventional pairs.
