What can be sown directly outdoors in May?

In our latitudes, May is a very special month in the gardening year as, at this time, the weather phenomena mix to a very special climate mix. Most often, spring already shows very clearly. Everything shines in the typical fresh green of the new sprouting of the plants, the fruit trees bloom in all their glory and there are those first, warm days that already give a taste of summer.
At the same time, there is the weather phenomenon of the so-called "Ice Saints" (also known as "icemen") that makes night frosts possible until mid-May. For that reason, sensitive plants like tomatoes, peppers, aubergines, and many more should only be planted outdoors after the Ice Saints.
Conversely, that also means that it's better to preculture such plants indoors to make a harvest possible that year. Especially peppers and aubergines have a very long maturation period so that the fruits may not mature fully if they are only sown after the Ice Saints.