Slice the carrots and the radishes thinly, preferably on a mandoline.
Toss the carrots into a skillet with the olive oil and some salt, pepper and fresh thyme. Sauté for about 4 minutes, until carrots are just starting to soften but don't lose their crunch.
Remove from heat and add black radish, letting the warm skillet just barely soften it. Let cool, and add red, white and purple radishes and toss in the oil to coat.
Combine dressing ingredients in a jar and shake to emulsify. Add carrot and radish mixture to the greens, drizzle with dressing and toss to lightly coat.