For the lentils: Set a large saucepan over high heat. Add the lentils and toast in the dry pan until they just turn in color a little (this will give the finished lentils a little texture), 20 to 30 seconds. Add 3 1/2 cups water, the cumin, red pepper flakes, bay leaf and some salt and pepper. Bring to a boil, and then reduce the heat and simmer, uncovered, until the lentils are tender and all the liquid has been absorbed, about 12 minutes. Pour the lentils out onto a flat platter in a thin layer so they cool faster and don't get too soft. Cool in the refrigerator for 5 minutes.
For the salad: Drain the giardiniera, reserving 1 tablespoon of the pickling juice, and roughly chop. Wash the arugula and dry well.
For the vinaigrette: In a large mixing bowl, make a quick vinaigrette by combining the reserved pickling juice with the red wine vinegar, Dijon mustard, sugar, and some salt and pepper. While whisking, add the olive oil to emulsify.
Assemble the salad in a large bowl by folding the giardiniera in with the lentils. Dress lightly with the vinaigrette and serve immediately over the arugula.