SCLEROTHERAPY vs. EVLT? I've obtained 2 consultations, one doctor recommends sclerotherapy and the other EVLT?

This needs microphlebectomy.Unfortunately, a lot of vein "specialists" are medical doctors, not surgeons, and they can only do one procedure (or two): endovenous ablation or sclerotherapy.Bulging knee veins need the underlying venous insufficiency with EVLT AND microphlebectomy by someone who has surgical training.Call us for our expertise. 
H Karamanoukian MD FACS
certified vein specialist (ABVLM) and double board certified surgeon

Varicose veins are usually due to malfunctioning valves of the saphenous system of veins-either the great, small or both.  A venous duplex ultrasound will determine this and treatment would be based on the ultrasound results.  If you have reflux then this should be treated otherwise the varicose veins will simply recur.  If no reflux (which I would doubt from your description) then treat only the varicose veins.

This will depend on the size and the tortuosity of the veins. Each surgeon will offer what they think is better for your veins in their experience. If you send some pictures then I might be able to assess your veins better

First and most important you would need to have dupplex ultrasound to identify circulation of saphenous veins. If you have leaking valves in saphenous veins, then it is probable cause of your varicose veins. In this case you would need to have EVLT of damaged saphenous vein +/- sclerotherapy of bulging veins. Bulging behind a knee is usually because Small saphenous vein damage.If, Otherwise, saphenous veins are normal, and the bulging vein is less than 4mm in diameter, sclerotherapy will work without EVLT.Make sure facilities you choose are accredited vascular labs and physician is ABVLM certified.