The whole game is about learning something about $\mu$, so you're right in your first question: the starting point is that $\mu$ is unknown and you want to use data and an estimator to learn about it. Using a given dataset, you can form an estimator $\hat \mu$ to estimate the value of $\mu$.
Formally, we usually think about $\mu$ as being a parameter (for instance, the expectation of a random variable, or the coefficient in a linear regression), and the estimator $\hat \mu$ as being a random variable, which depends on the realisation of the data. For a given dataset, you will obtain a given estimate.
Let us assume (as you do) that $\hat \mu$ is normally distributed. Its expectation is equal to $\mu$ (which happens when the estimator is consistent). Expectation, in this case, means that if you were to observe not one dataset, but a large number of datasets, the average value of the $\hat \mu$ over these datasets would be equal to $\mu$.
The variance of $\hat \mu$ is a function of two quantities: $\sigma$ the asymptotic/underlying variance, and $n$ the number of observations. $\sigma$ essentially depends on the data generating process of the random variable (for instance, the variance of the underlying random variable). Variance of $\hat \mu$ means: how would my $\hat \mu$ vary if I computed it many times, on many datasets of size $n$? If $n$ was very large, all the $\hat \mu$ would be pretty close to each other (and pretty close to $\mu$).
Now that we have all this, let's answer your question. You're right that the formula of the variance (and CI) of $\hat \mu$ depends on $\sigma$ and we don't observe $\sigma$. What is usually done is to plug an estimator of $\sigma$ instead. In many cases, just like you can form an estimator of $\mu$, you can form an estimator of $\sigma$, that you call $\hat \sigma$, which can be computed as a function of the data.
For instance, the canonical problem is that you have a random variable $Y_i$ distributed in a $\mathcal N(\mu, \sigma^2)$. In this case, we can take $\hat \mu$ to be the average of the observed $Y_i$. An unbiased estimator of the variance $\sigma^2$ is then:
$$
\hat \sigma = \frac{\sum_i Y_i^2}{n} - \hat \mu^2
$$
Note that $\hat \sigma$ depends on $n$ and on the observations of the dataset $\{Y_i\}$ and on the estimator $\hat \mu$ (which also depends on $\{Y_i\}$ and $n$), but not on $\mu$. $\hat \sigma$ is the quantity you will plug into your confidence interval.