Is there a way to determine which face of a part the mouse is hovering over?

For instance, my mouse.

If I hover over the back of the part, I should be able to determine it is the back. Any suggestions?

PlayerMouse.TargetSurface is exactly what you want!

--LocalScript

local mouse = game.Players.LocalPlayer:GetMouse()

mouse.Move:connect(function()
	if mouse.Target then
		print(mouse.TargetSurface.Name)
	end
end

If you're not used to working with EnumItems, there is a page for them on the wiki as well.