Let me summarise the implications of [[Blecher–Kaad–Mesland 2018][1], Section 3.4]. I'll assume that $A$ and $B$ are unital $C^\ast$-algebras.

 1. Let $\mathcal{A} := \{a \in A \mid [D,a] \in B(H)\}$ endowed the so-called Lipschitz norm $\|a\|_D := \|a\| + \|[D,a]\|$. Then the inclusion $\mathcal{A} \hookrightarrow A$ is contractive with dense range closed under the holomorphic functional calculus (see [[Mesland 2012][2], Section 4.2]). For convenience, let $\Omega_D$ denote the unital $C^\ast$-subalgebra generated by $A$ and $[D,\mathcal{A}]$, so that the inclusion $A \hookrightarrow \Omega_D$ makes $\Omega_D$ into a Hilbert $(A,\Omega_D)$-correspondence.

 2. Let $\nabla : \mathcal{E} \to E \hat\otimes_A \Omega_D$ be a *Hermitian connection*, i.e., a $\mathbb{C}$-linear map with domain a dense subspace $\mathcal{E}$ of $E$ satisfying $\mathcal{E} \cdot \mathcal{A} \subset \mathcal{E}$ and $(\mathcal{E},\mathcal{E})_A \subset \mathcal{A}$, such that $$\forall e \in \mathcal{E}, \, \forall a \in \mathcal{A}, \quad \nabla(ea) = \nabla(e)a + e \hat\otimes [D,a],\\ \forall e_1,e_2 \in \mathcal{E}, \quad (e_1 \hat\otimes 1,\nabla(e_2))_{\Omega_D} - (\nabla(e_1),e_2 \hat\otimes 1)_{\Omega_D} = [D,(e_1,e_2)_A].$$ Then $\nabla$ is, in fact, a closeable operator, and the domain $\mathcal{E}_\nabla$ of the closure satisfies $\mathcal{E}_\nabla \cdot \mathcal{A} \subset \mathcal{E}_\nabla$ and $(\mathcal{E}_\nabla,\mathcal{E}_\nabla)_A \subset \mathcal{A}$.

 3. Now, for any $b \in B$, such that $b \cdot \mathcal{E}_\nabla \subset \mathcal{E}_\nabla$, we can define $\delta_\nabla(b) : \mathcal{E}_\nabla \otimes^{\mathrm{alg}}_\mathcal{A} \Omega_D \to E \hat\otimes_A \Omega_D$ by $$\forall e \in \mathcal{E}_\nabla, \, \forall \omega \in \Omega_D, \quad \delta_\nabla(b)(e \hat\otimes \omega) := \nabla(be)\omega - (b \hat\otimes 1)\nabla(e)\omega.$$ Now, let $$\mathcal{B} := \{b \in B \mid b \cdot \mathcal{E}_\nabla + b^\ast \cdot \mathcal{E}_\nabla \subset \mathcal{E}_\nabla, \; \delta_\nabla(b),\delta_\nabla(b^\ast) \in B(E \hat\otimes_A \Omega_D)\}, $$
and observe that $\mathcal{B}$ is dense in $B$ since it contains the ket-bra $\lvert e_1 \rangle \langle e_2 \rvert$ for every $e_1,e_2 \in \mathcal{E}_\nabla$. It turns out that $\delta_\nabla : \mathcal{B} \to \operatorname{End}_{\Omega_D}(E \hat\otimes_A \Omega_D)$ is a closed $\ast$-derivation densely defined on $B$; indeed, if we endow $\mathcal{B}$ with the Lipschitz norm $\|b\|_{\nabla} := \|b\| + \|\delta_\nabla(b)\|$, then the inclusion $\mathcal{B} \hookrightarrow B$ is contractive with dense range closed under the holomorphic functional calculus.

 4. At last, define $1 \hat\otimes_\nabla D : \mathcal{E}_\nabla \otimes^{\mathrm{alg}}_{\mathcal{A}} \operatorname{Dom} D \to E \hat\otimes_A H$ by
$$
 \forall e \in \mathcal{E}_\nabla, \, \forall h \in \operatorname{Dom}(D), \quad 1 \hat\otimes_\nabla D(e \hat\otimes h) := \nabla(e)h + e \hat\otimes Dh;
$$
observe, then, that for any $b \in \mathcal{B}$,
$$
 \forall e \in \mathcal{E}_\nabla, \, \forall h \in \operatorname{Dom}(D), \quad [1 \hat\otimes_\nabla D,b \hat\otimes 1](be \hat\otimes h) = \delta_\nabla(b)(e \hat\otimes 1)h.
$$
The methods of [[Brain–Mesland–Van Suijlekom][3] 2016] now suffice to show, in particular, that $1 \hat\otimes_\nabla D$ is essentially self-adjoint, yielding a spectral triple $(B,E \hat\otimes_A H,1 \hat\otimes_\nabla D)$ with $\mathcal{B} = \{b \in B \mid [1 \hat\otimes_\nabla D,b] \in B(E \hat\otimes_A H)\}$, precisely since $[1 \hat\otimes_\nabla D,\cdot]$ is essentially the same thing as $\delta_\nabla$.

Finally, note that Hermitian connections on $E$ actually do exist. If $E \cong p_0 A^N$ for some orthogonal projection $p_0 \in M_N(A)$, one can use the fact that $\mathcal{A}$ is closed under the holomorphic functional calculus to find an orthogonal projection $p \in M_N(\mathcal{A})$, such that $E \cong p A^N$, in which case, one can take $\mathcal{E} := p \mathcal{A}^N$ and define $\nabla$ by
$$
 \nabla(e) := \sum_{i=1}^N \xi_i \hat\otimes [D,(\xi_i,e)_A]
$$
where $\{\xi_1,\dotsc,\xi_N\} \subset \mathcal{A}^N$ are the columns of $p$; this is the so-called *Graßmann connection* induced by the projection $p$, which is the default connection in work focussed more strictly on index theory. If I recall correctly, in this case, you get $\mathcal{E}_\nabla = \mathcal{E}$ and $\mathcal{B} = p M_N(\mathcal{A}) p$.


  [1]: https://arxiv.org/abs/1703.10063
  [2]: https://arxiv.org/abs/0904.4383
  [3]: https://arxiv.org/abs/1306.1951