Brute force might work

The thought behind this is that if an even integer $n$ cannot be expressed as a sum of two composite numbers, then for every odd composite number $0 < k < n$ , $n - k$ is non-composite, meaning that it is $1$ or prime. Now, we just need to find $n$ such that there exist an odd prime $p$ such that there exist an odd composite number $k < n - p$ of each remainder class of $p$ meaning that one of $n - k$ 's is divisible by $p$ and greater than $p$ so $n = k + n - k$ can clearly be expressed as sum of two odd composite numbers.

Here we start checking case $p = 3$ and after we find $35, 27$ and $25$ we have our upper bound of $38$ which after checking is found to be the solution.