What is the synoptic/atmospheric setup that is causing the U.S. severe weather outbreak in May 2019?
Summarizing the comments I made above in this question and in this one - Does this weather pattern have a name ? I believe significant parts of the US  are experiencing a  Stationary Front for the past one month. There has been plenty of media coverage of this event and if you google the term "stationary front" under news one will receive a lot of links to the TV news media with meteorologists giving detailed analysis of this event. 

A few summarized below - 

1) Week's bleak weather not an unusual spring occurrence 

2)A stationary front is stuck across the middle parts of Illinois and Indiana

I will attempt to provide a synoptic scale explanation of the event from reanalysis data. 

1) From JMA's site  -  

an upper level look at the 200 hPa surface reveals that an equatorward trajectory of the wave activity flux vector(seen as small arrows) is seen  from the Upper North Pacific. Climatologically Rossby Wave Breaking can occur from boreal fall to boreal spring. The signal is seen at 30 day , 10 day period and 7 day period and 5 day period. The signal is present in the raw data as well as in the anomaly. Most importantly this wave activity flux is directed towards the continental USA all the way from New Mexico to the Great Lakes region. 

A popular science version of the above information has been discussed over at this link - What's up with all this wet weird weather ?

2) At the surface level this "stationary front" presents itself as air masses of two different temperatures as seen from this plot and one can see the frontal boundary more or less coincides with the rising contour of the stream function plot in (1). Here the south easterlies are bringing in warm moist air from the Gulf coast and the Atlantic and the colder air masses are coming from Canada and North East Pacific. So the key question is whether the upper level influence is directing the surface level weather and that question can be answered by calculating PV anomalies when ERA 5 reanalysis data does become available. 



Source - JMA reanalysis data

3) The moisture flux being pumped into the south eastern part of the USA can be seen from this plot over a seven day period - .
Again the signal is present over 30 day period, 10 day period and 3 day period and over a 1 day period. I would like to point out the moisture flux is seen at different levels of the atmosphere from surface level to the 300 hPa surface. 

Source - NCAR NOAA reanalysis derived variables

4) Last year (2018) apparently a wavenumber - 7 was found to be causing weather stalls as shown in this link - Weather stalls becoming longer. This year in April  it is a wavenumber 5 pattern as seen in this site - Current Weather reports and summarized here


  From a global point of view, the weather pattern in the beginning parts of the 6-10 day (around day 7) will enter into a wavenumber 5 pattern. That's where you have 5 distinct upper-level troughs (areas where the weather is cool and stormy) around the globe. It's like a major traffic jam in the atmosphere and could lead to weather stagnation with little to no variation for days and/or weeks. This could also lead to extreme weather events such as droughts and severe weather/flooding across some parts of the world.


and I am able to pick up the wavenumber 5 or 6 as pointed by user Deditos (as well in this twitter feed - Wave number 6 planetary wave) from NCAR NOAA reanalysis data for the 500 hPa geopotential height anomaly for the month of May.