This Berceuse by Russell King Miller is a gentle character piece reflective of the late Romantic parlor style. It is distinguished by its swaying melodic contour and smooth, cradling harmonic accompaniment.
The primary challenge is maintaining a consistent, hypnotic pulse while balancing the melody against a delicate left-hand accompaniment.
Comparable difficulty and character — good candidates for what to learn next.
Berceuse (Op.16 No.1) is rated Henle level 3 of 9 — late beginner repertoire, roughly ABRSM grades 4–5. In the RCM system it corresponds to about level 4.
You should be comfortable at Henle level 3 — meaning you can already play level 2–3 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 (Op.16 No.1) lasts about 3 minutes.
Berceuse (Op.16 No.1) 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 Au Matin by Benjamin Godard, Au Matin by Benjamin Louis Paul Godard, Au Matin by Benjamin Louis Paul Godard — see the full similar-pieces list on this page.