There is a generalisation of the character formula, although it is usually stated in terms of contents rather than Frobenius coordinates. Also, it applies to conjugacy classes labelled by partitions $(1^{m_1} 2^{m_2} \cdots)$, where $m_2, m_3, \ldots$ are fixed and $m_1$ varies with $n$ (as in the example you gave, where there is one part of size $2$, and $n-2$ parts of size 1). So in particular, it does not apply directly to the partitions $(2^{n/2})$.

For the benefit of the uninitiated, let me state right away that $\frac{|C_\mu| \chi_{\mu}^\lambda}{dim(\lambda)}$ is a *central character*. The sum of all elements of $S_n$ of cycle type $\mu$ is a central element of the group algebra of $S_n$, and the central character is the scalar by which the central element acts on the irreducible representation labelled by $\lambda$. Central characters are important in modular representation theory, which might provide a little motivation for why the quantity in question is a natural thing to consider.

Suppose that we have a partition $\lambda = (\lambda_1, \ldots, \lambda_l)$. Suppose that $d$ is the largest integer such that $\lambda_d \geq d$ (the *Durfee size* of $\lambda$). Then the Frobenius coordinates of $\lambda$ are $(\alpha_1, \ldots, \alpha_d | \beta_1, \ldots, \beta_d)$, where $\alpha_i = \lambda_i - i$ and $\beta_i = \lambda_i^\prime - i$ (here $\lambda^\prime$ is the transpose partition of $\lambda$). So the $\alpha_i$ and $\beta_i$ count the number of boxes in past the diagonal in the $i$-th row and $i$-th column respectively (and the number of each is $d$). On the other hand, the contents of a box in row $j$ and column $i$ is $i-j$. It is not difficult to see that the multiset of contents of $\lambda$ consists of $0$ with multiplicity $d$, and $\{1,2,\ldots, \alpha_i\}$ for each $i$ together with $\{-1, -2, \ldots, -\beta_i\}$ for each $i$. So in particular, the sum of $r$-th powers of the contents of $\lambda$ can be expressed as polynomials in the $\alpha_i$ and $\beta_i$ by Faulhaber's Formula. For example, the sum of the contents ($r=1$) is

$$\sum_i (1 + 2 + \cdots + \alpha_i) - (1 + 2 + \cdots + \beta_i) = \sum_i {\alpha_i + 1 \choose 2} - {\beta_i + 1 \choose 2},$$

which is precisely the quantity appearing in the example you gave. We write $cont(\lambda)$ for the multiset of contents of $\lambda$.

Now, for a partition $\mu = (\mu_1, \mu_2, \cdots, \mu_k)$, let $\bar{\mu} = (\mu_1 - 1, \mu_2 - 1, \ldots, \mu_k - 1)$ be the partition obtained from $\mu$ by subtracting 1 from each nonzero part of $\mu$ (and ignoring trailing zeros). For example, if $\mu = (2,1, \ldots, 1)$ as in your example, then $\bar{\mu} = (1)$.

**Theorem** There exists a family $f_{\rho}$ of elements of $\mathbb{Q}[n] \otimes \Lambda$ (where $n$ is a polynomial variable) and $\Lambda$ is the ring of symmetric functions with the property that

$$\frac{|C_\mu| \chi_{\mu}^\lambda}{dim(\lambda)} = f_{\bar{\mu}}(|\lambda|, cont(\lambda))$$

By this notation I mean that we evaluate $f_{\bar{\mu}}$ at $n=|\lambda|$ and its symmetric function variables at $cont(\lambda)$.

It turns out that $f_{(1)} = p_1$ (the first power-sum symmetric function) which doesn't actually depend on $n$. Then putting this together, we see

$$\frac{|C_{(2,1,\ldots,1)}| \chi_{(2,1,\ldots,1)}^\lambda}{dim(\lambda)} = p_1(cont(\lambda)) = \sum_i {\alpha_i + 1 \choose 2} - {\beta_i + 1 \choose 2},$$

which recovers the result you stated. In general, you can take $f_{\bar{\mu}}$, express it as a polynomial in $n$ and the power-sums $p_1, p_2, \ldots \in \Lambda$, and upon evaluating at the contents of $\lambda$ you will obtain a polynomial in $n$ and the Frobenius coordinates of $\lambda$. For example,

$$f_{(2)} = p_2 - {n \choose 2},$$

which tells us that the central character corresponding to the conjugacy class $(3,1,\ldots, 1)$ may be computed using the formula

$$\left( \sum_{i} \frac{\alpha_i(\alpha_i + 1)(2\alpha_i + 1)}{6} + \frac{\beta_i (\beta_i + 1)(2\beta_i + 1)}{6} \right) - {|\lambda| \choose 2}.$$

There is a lot more to be said about the $f_\rho$. For example, they form a $\mathcal{R}$-basis of $\mathcal{R} \otimes \Lambda$, wher $\mathcal{R}$ is the subring of $\mathbb{Q}[n]$ consisting of integer-valued polynomials. But in this post I worked over $\mathbb{Q}$ because I wanted to use power-sum symmetric functions to turn contents-evaluaion into polynomials in Frobenius coordinates.

References you might find helpful (each of which contains an explanation of the above theorem):

*Corteel, Sylvie; Goupil, Alain; Schaeffer, Gilles*, **Content evaluation and class symmetric functions**, Adv. Math. 188, No. 2, 315-336 (2004). ZBL1059.05104.

Section 5.4 of

*Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo*, Representation theory of the symmetric groups. The Okounkov-Vershik approach, character formulas, and partition algebras., Cambridge Studies in Advanced Mathematics 121. Cambridge: Cambridge University Press (ISBN 978-0-521-11817-0/hbk). xv, 412 p. (2010). ZBL1230.20002.

(although this is based on the material of the above paper)

Section 3 of my own paper (which doesn't appear in the citation engine, so I'm providing an arXiv link instead) Stable Centres I: Wreath Products