In a large saucepan of boiling salted water, blanch the carrots for 2 minutes. Using a slotted spoon, transfer to a colander and refresh under cold water; drain well. Repeat with the fennel and broccoli rabe.
In a small saucepan, melt the butter in 1/4 cup plus 2 tablespoons of the olive oil. Add the anchovies and cook over low heat, stirring, for 4 minutes. Add the minced garlic and 1/2 teaspoon of the thyme and cook until the garlic is golden, about 3 minutes. Add the lemon juice and season with salt and pepper. Remove the dressing from the heat.
Heat the remaining 2 tablespoons of olive oil in a large skillet. Add the onion, the remaining 1/2 teaspoon of thyme and the chile and cook over moderately high heat just until the onion softens, about 2 minutes. Add the fennel and broccoli rabe and cook, stirring, just until heated through, about 3 minutes. Stir in the dressing. Transfer the vegetables to a large shallow bowl. Add the carrots, endives and radishes, season with salt and pepper and toss well and serve.
A fresh, light and round white with soft acidity and no oak will balance the saltiness of this salad with its fresh vegetal character.