Cook broccoli in a large pot of boiling salted water until bright green and crisp-tender, about 2 minutes. Drain; transfer to a bowl of ice water. Let cool, then drain. Pat broccoli dry; place in a large bowl.

Prepare a grill for medium-high heat. Drizzle broccoli with oil and toss to coat; season with salt. Grill broccoli, turning occasionally, until charred in spots, 5-7 minutes. Return to bowl and let cool, tossing occasionally.

Purée anchovy, garlic, mayonnaise, buttermilk, chervil, tarragon, chives, lemon juice, and mustard in a blender until smooth; season dressing with salt.

Add arugula to broccoli and toss to combine. Drizzle salad with 3/4 cup dressing and toss to coat; season with more salt if needed.

Arrange salad on a platter and top with cheddar.