This Berceuse by Peterson-Berger is a gentle character piece from his 'Frösöblomster' collection. It is distinguished by its Nordic harmonic flavor and intimate, lullaby-like melody.
The primary challenge is maintaining a consistent, rocking left-hand accompaniment while balancing the melodic line.
Comparable difficulty and character — good candidates for what to learn next.
Berceuse is rated Henle level 4 of 9 — intermediate repertoire, roughly ABRSM grades 5–6. In the RCM system it corresponds to about level 6.
You should be comfortable at Henle level 4 — meaning you can already play level 3–4 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of Berceuse lasts about 3 minutes.
Berceuse is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Nocturne in E-flat major, No. 2 by Frédéric Chopin, Au Matin by Benjamin Louis Paul Godard, Au Matin by Benjamin Louis Paul Godard — see the full similar-pieces list on this page.