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 ¾ cup dressing and toss to coat; season with more salt if needed.
Arrange salad on a platter and top with cheddar.
Do Ahead: Dressing can be made 2 days ahead. Cover and chill.
Thank you to https://www.bonappetit.com/recipe/grilled-broccoli-and-arugula-salad for this awesome recipe!