According to

Lev Kazarin, On Thompson’s Theorem,

Journal of Algebra220, 574–590 (1999)

we know that:

[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$ where $$f_1\lt \dots\lt f_n,\;\;\; f_1|f_2|\dots|f_n,\;\;\;(d,f_n)=1.$$ If $Irr(G)$ contains only one character of degree $d$, then the following assertions hold:

(a) $G$ contains a normal subgroup $N$ such that $G/N$ is cyclic of order $d$.

(b) $cd(N)=\{1,f_1,\dots,f_n\}$ so that $N$ has an ordered Sylow tower.

Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?