Bring a large pot of water to a boil and add a liberal amount, 2-3 tablespoons, of salt (no, this will not all go into your food, but it will season everything you cook in it. Trust me.) Add the potatoes to the water and boil until just tender, but not mushy. Remove them to a colander, rinse with cool water and then dice into cubes and add to a large bowl. Meanwhile add in the carrots, boil for 2-3 minutes until tender but still crisp. Remove them to the colander, rinse with cold water, add them to the potatoes. Continue with the lima beans, which should take about 5 minutes, and then to the peas, which should only need a minute or two. Rinse both under cold water and add to the potatoes and carrots. Throw the artichokes in the water to heat up and remove any tinny taste and then add to the rest of the vegetables.
While your vegetables are blanching, beat the egg yolks until foamy, then while whisking, very slowly drizzle in the oil (mix the two oils together in a measuring cup first). The mixture should be very thick (like mayo!). Once all of the oil is absorbed stir in the garlic ,tarragon, half the lemon juice and season with salt. Add the mayo to the vegetables and gently stir to combine. Season with pepper and additional salt and lemon juice if needed. Refrigerate for a couple of hours or up to 3 days.