HAML

.natural-curves
    .natural-curves-inner
        …

Let your bordered outer div determine the background1 and its border-radius2 will put a natural curve on the inner div.

If you set the background of the inner div, you will also need to set its border-radius3 and you will lose your natural curves.

CSS

.natural-curves {
    border: 0.5em solid;
    border-radius: 1.5em; /* 2 */
    background: lightblue; /* 1 */
    …
}
.natural-curves-inner {
    overflow: hidden;

/* 3 */
/*
    border-radius: 1em;
    background: burlywood;

*/
}