In the key of C, what type of cadence and Roman numeral analysis would G B D# to C E G be? It's in John Thompson, "Hop O' My Thumb" measures 3 and 4.
In this case, the dominant is just an augmented triad, so the Roman numeral would be V+. And since it's V moving to I, it's still going to be an authentic cadence.
One of the reasons this progression works so well is the half-step resolution from both B to C and from D♯ to E. In fact, these V+ chords usually start as regular V triads that then have chromatic motion from (in this case) D to D♯ to make it a V+. This is exactly what happens on "keep you satisfied" in The Beatles' "From Me to You."
You'll also notice that G, B, and D♯ are exactly half of a whole-tone collection. Beginning around the end of the 19th century some composers would alter dominants so much that they were basically just whole-tone sets. A V+(♯11), for instance, is an entire whole-tone collection: G B D♯ F A C♯.
Sharp 5 is common enough in dominant chords that I would call this an authentic cadence, from a dominant V to the tonic I. The sharp 5 (D#) in the G chord increases the tension, which fits with a dominant function chord.