$\newcommand{\E}{\mathbb{E}}$
I have someone who writes expectation as being $\E(X - \E(X))^2$
Is it possible for this to be notationally correct?
I would expect to rather read $\E((X - \E(X))^2)$ for instance, if not $\E[(X - \E(X))^2]$
I've encountered this in a university lecture, but I was wondering if there were an educational tradition in some probability texts which would see this written in this way.
The expectation of a random variable is defined via an integral and integration is linear, hence expectation is linear—indeed, it is of a class of objects called linear operators. That is $$ \mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]. $$ When we study linear operators, we typically omit the parentheses.
As a basic example, think about linear algebra. An $n\times n$ matrix $A$ is a way of representing a linear operator. That is, we can think of $A$ as a function $$ A : \mathbb{R}^n \to \mathbb{R}^n, $$ which acts on elements of $\mathbb{R}^n$ via matrix multiplication. If $x\in\mathbb{R}^n$, we can think of $A$ a function acting on $x$ and write $A(x)$ or $A[x]$, or we can think of $A$ as a matrix that acts by matrix multiplication on the column vector $x$ and write $Ax$. The notation here is unambiguous, so we typically omit the braces.
We adopt a similar approach to notation when we deal with linear operators in general. That is, if $T : X \to Y$ is a linear operator ($T(x+y) = T(x) + T(y)$ for all $x$ and $y$, plus the necessary hypotheses on the spaces $X$ and $Y$ (basically, they are topological vector spaces), plus some other technical conditions), then we typically write $Tx$ rather than $T(x)$.
Because expectation is linear, it is reasonable to adopt the same notation, and write $\mathbb{E}X$ rather than $\mathbb{E}[X]$ or $\mathbb{E}(X)$, which is often done. That is (to finally answer your question), it is common to see notation such as $$ \mathbb{E}(X-\mathbb{E}X)^2 \qquad\text{or}\qquad \mathbb{E}(X-\mathbb{E}(X))^2. $$ Because it is possible that this notation could be ambiguous, one could be forgiven (or even lauded!) for more explicitly writing $$ \mathbb{E}\left[ (X-\mathbb{E}(X))^2 \right]. $$ On the other hand, in the particular example that you have given, the expression gives the variance; that is $$ \mathbb{E}(X-\mathbb{E}X)^2 =: \operatorname{Var}(X). $$ Because this is such a common expression, it is even more reasonable to assume that the reader is familiar with the notation, and is willing to accept some slightly more terse notation.