We construct $Δ$-operators $F[Δ]$ on the space of almost symmetric functions $\mathscr{P}_{as}^{+}$. These operators extend the usual $Δ$-operators on the space of symmetric functions $Λ\subset \mathscr{P}_{as}^{+}$ central to Macdonald theory. The $F[Δ]$ operators are constructed as certain limits of symmetric functions in the Cherednik operators $Y_i$ and act diagonally on the stable-limit non-symmetric Macdonald functions $\widetilde{E}_{(μ|λ)}(x_1,x_2,\ldots;q,t).$ Using properties of Ion-Wu limits, we are able to compute commutation relations for the $Δ$-operators $F[Δ]$ and many of the other operators on $\mathscr{P}_{as}^{+}$ introduced by Ion-Wu. Using these relations we show that there is an action of $\mathbb{B}_{q,t}^{\text{ext}}$ on almost symmetric functions which we show is isomorphic to the polynomial representation of $\mathbb{B}_{q,t}^{\text{ext}}$ constructed by González-Gorsky-Simental.