Flights from Austin to JFK are relatively inexpensive. Now, the Skyteam airline alliance (Delta, Air France, Air Europa, Alitalia, KLM) are offering airfare of just 1085 Euros for the long trans-Atlantic route in comfortable Business Class.
Aer Lingus, Delta, Air France, Lufthansa, Air Canada, and United are offering some low airfares from DFW (and other American cities) to Dublin, Ireland. A variety of date pairs (with varying airfares) have been found from March to June, and August through December.
Delta/KLM/Air France are offering flights from Austin to Belgium, Germany and Switzerland for as little as $514 round-trip. You can travel with these low, low fares from Austin in March, April, and August to December.
Momondo lists the same flights as low as $488.
You may find these flights listed on the Air France website, which may be codeshared with Delta.
Scotts Cheap Flights has found some round-trip airfares from Houston to Europe in the $400 range. Travel up through May 2018, with some also in August 2018. Mostly Delta, some Air France and Air Canada.
European destinations from Houston include Brussels, Frankfurt, Geneva and Munich. Of course, you can take a European UCC (Ultra low-cost carrier, like Ryanair or Easy Jet) from any of those cities to wherever you want to go in Europe.
Other departure cities are also available, including Chicago (take Southwest or American non-stop from Austin to Chicago), Denver, Las Vegas and Washington, D.C.