Adjust an oven rack to the lowest position, place a large rimmed baking sheet on the rack, and heat the oven to 500°F. This intense bottom heat is essential for browning the vegetables without overcooking them.

Toss the fennel and mushrooms with oil, salt, pepper, and sugar in a large bowl. Working quickly to keep the oven temperature high, carefully spread the mixture over the hot baking sheet in an even layer.

Roast until the vegetables are tender and well browned on the bottom, about 20 to 25 minutes. Do not stir during roasting; the uninterrupted contact with the hot metal is what creates the deep, sweet caramelization.

Whisk the lemon juice, mustard, salt, and pepper together in the bowl used for the vegetables. While whisking constantly, slowly drizzle in the remaining oil until the dressing is thick and emulsified.

Add the hot roasted vegetables to the bowl with the vinaigrette and toss to coat. Let the mixture cool to room temperature, about 30 minutes, allowing the fennel to absorb the acid from the lemon.

Stir in the radishes and transfer the salad to a serving platter. Sprinkle with the reserved fennel fronds and serve while still slightly warm or at room temperature.