Bring 4 quarts of water to a boil and add 2 tablespoons of salt. Fill a large bowl with ice cubes and cold water, and place it next to the stove. Cook the agretti in the boiling water until just barely tender, about 2 minutes. Remove the agretti and plunge into the ice water. When they have cooled, drain the agretti again and spread on paper towels to dry.
In the meantime, combine mustard, capers, fennel fronds, hard-boiled egg and white wine vinegar in a large bowl. Slowly whisk in the olive oil. Toss agretti with the vinaigrette.
In another large bowl, combine the shaved fennel with the anchovy vinaigrette and toss until the fennel is coated. Divided between four plates: place the fennel salad next to the dressed agretti, garnish with shaved Cacio di Roma, drizzle with a bit more caper vinaigrette, and serve immediately.
In a 12- to 14-inch sauté pan, simmer the anchovies, garlic and oil over low heat for 30 minutes, or until the anchovies are falling apart. Remove from heat, strain out the solids, and chill the oil overnight in a covered container. Place the chilled oil in a food processor. Add the lemon juice and vinegar and blend until smooth. Adjust the seasoning with salt and pepper.