Blanch the vegetables: Bring a large pot of salted water to a boil. Fill a large bowl with ice water. Boil the Brussels sprouts until bright green and just tender, about 7 minutes. Remove with a slotted spoon and transfer to the ice water. Once cool, remove and pat dry. Repeat to blanch the radishes, cooking until just tender, 5 to 7 minutes.
Heat 1 tablespoon butter in a large skillet over medium heat. Add the shallot and cook until softened, 3 to 4 minutes. Add the Brussels sprouts and radishes, season with salt and pepper and cook, stirring, until heated through, 3 to 5 minutes. Add 1 tablespoon water, then gently stir in the remaining 3 tablespoons butter, 1 tablespoon at a time, until melted. Stir in the tarragon and parsley.
Blanch the vegetables (step 1) up to 1 day ahead; cover and refrigerate.