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As you are bringing a cycle and panniers, etc, you may find flying to be the cheapest and most convenient. Because Thailand is a popular destination for European tourists there are usually fare deals being offered.
But there is no etched in stone cheapest routing, the fares vary from day to day and so you need to sit down and do some searching on the internet AND check the Sunday travel sections.
There are a myriad of routings, direct Europe to BKK flights, connecting via the Middle East, connecting via Russia or China, connecting via South Asia. While Bangkok is the main airport to check, you might also look at Phuket, Kuala Lumpur as well for the long haul destination. Cities in Laos have very few international connectors.
And don't use maps you bought at home to guide your cycling, pick up some maps from ThinkNet when you get here. Can't begin to count the number of self-contained riders I have encountered riding up the six lane highways because that is all their international maps shows, while there is a beautiful two lane rural highway paralleling just 1 kilometer over.