Remove any damaged outer leaves from the radicchio, quarter it, remove the core, and cut each quarter crosswise into 1-inch widths.
Position a rack in the center of the oven and heat the oven to 350°F.
Put the breadcrumbs on a rimmed baking sheet and toss lightly with 2 Tbs. of olive oil to coat evenly. Spread in a thin, even layer and season lightly with salt. Bake, stirring every few minutes, until crisp and light golden brown, about 10 minutes. Set aside to cool on the pan.
Bring a large pot of well-salted water to a boil over high heat. Cook the cauliflower in the boiling water until just tender, about 3 minutes. Drain, spread on a baking sheet, and set aside at room temperature.
Put the garlic in a mortar, add a pinch of salt, and pound to a paste with a pestle. Or mince and then mash to a paste with the side of a chef’s knife. Combine the garlic, vinegar, lemon juice, and mustard in a small bowl. Let sit for 5 to 10 minutes. Whisk in the egg yolk, and then slowly whisk in the remaining olive oil until emulsified. Taste with a piece of cauliflower and season with more vinegar or salt if necessary.
Put the cauliflower in a large bowl and season with salt and pepper. Gently toss with just enough vinaigrette to lightly coat. Add the radicchio and parsley, season with salt and pepper, and toss again with just enough vinaigrette to lightly coat. Taste and add more salt or vinaigrette if necessary. Gently transfer the salad to a platter or individual serving plates; it’s best if it’s arranged somewhat flat. Drizzle any remaining vinaigrette on the salad (you may not need it all), scatter the breadcrumbs on top, and serve.
For a tasty variation, try adding black olives or finely chopped hard-cooked egg. To make the salad a meal, serve thinly sliced, seared skirt steak alongside.