  We report on high-precision $Q_{\textrm{EC}}$ values of the
$^{21}$Na$\rightarrow^{21}$Ne and $^{23}$Mg$\rightarrow^{23}$Na mirror
$\beta$-transitions from mass measurements with ISOLTRAP at ISOLDE/CERN. A
precision of $\delta m/m = 9 \cdot 10^{-10}$ and $\delta m/m = 1.5 \cdot
10^{-9}$ was reached for the masses of $^{21}$Na and $^{23}$Mg, respectively.
We reduce the uncertainty of the $Q_{\textrm{EC}}$ values by a factor five,
making them the most precise experimental input data for the calculation of the
corrected $\mathcal{F} t$-value of these mixed Fermi/Gamow-Teller transitions.
For the $^{21}$Na$\rightarrow^{21}$Ne $Q_{\textrm{EC}}$ value, a $2.3 \sigma$
deviation from the literature $Q_{\textrm{EC}}$-value was found.
