Adjust an oven rack to the lowest position, place a large rimmed baking sheet on the rack, and heat the oven to 500°F. Placing the sheet in the oven while it preheats ensures the vegetables start searing the moment they hit the metal.

Toss the fennel and mushrooms with oil, salt, pepper, and sugar in a large bowl. Working quickly to keep the pan hot, carefully spread the vegetables over the hot baking sheet in an even layer. Roast until the vegetables are tender and well-browned on one side, 20 to 25 minutes—do not stir during roasting or you will lose that concentrated caramelization.

Whisk the lemon juice, mustard, salt, and pepper together in the bowl used earlier. Gradually drizzle in the remaining oil while whisking constantly until the dressing is smooth and emulsified.

Add the hot vegetables to the bowl with the vinaigrette and toss to coat, then let the salad cool to room temperature, about 30 minutes. This resting period allows the warm vegetables to soak up the bright flavors of the lemon and mustard.

Stir in the sliced radishes and transfer the salad to a serving platter. Sprinkle with the reserved fennel fronds and serve at room temperature.