Preheat your water bath to 135°F (57°C).
Season lamb and shape into a patty (I like about 1 - 1 1/4" thick), wrap in plastic wrap, and freeze for 1 hour or until it will hold its shape under vacuum.
Vacuum seal, then put into the bath making sure it's fully submerged.
Cook 2 - 4 hours (or until pasteurized based on your temp and thickness).
Combine all the ingredients in a measuring cup that just barely fits the width of your immersion blender. Make sure the oil is the last thing to go in.
Insert your immersion blender and rest on the bottom of the cup. Blend until a thick emulsion is obtained.
NOTE: You may need to move the blender up and down to make sure all the oil is incorporated, but be careful - you don't want to shoot aioli out of the cup. This can be refrigerated for 2-3 days in an airtight container.
Preheat your grill (or cast iron pan) as hot as you can get it. Remove the burger patty from the bag and dry well.
Sear patty on the first side for 30 - 45 seconds. If you want, you can use a torch to brown the sides while the bottom sears.
After 30-45 seconds, flip the patty and immediately add your cheese. Sear for another 30 - 45 seconds or until nicely browned.
To assemble, put a thin layer of the aioli on both sides of the bun. Place arugula on the bottom bun then add the patty, a slice of heirloom tomato, and the top bun.