Add the egg to a small bowl and beat until combined. Add the flour to another small bowl. In a small saute pan, set over medium-high heat, add the vegetable oil. Separate shallot—you should have like little baby onion rings! Line a plate with paper towels (you’ll use this to drain the shallots after frying). When the oil is hot, dip the shallot rings, in batches, into the egg mixture and then immediately in the flour. Drop them, carefully, into the hot oil and fry on each side for about 1 minute, until golden brown. Transfer to the paper towel to drain and immediately sprinkle them with a bit of salt. Repeat until you’ve worked your way through all of the shallot rings. Set aside.

Preheat the oven to 400 degrees F. Line a baking sheet with foil or parchment. Spread the broccolini out onto the baking sheet and drizzle with about a tablespoon of olive oil. Toss it together until it’s evenly coated. Transfer to the oven to roast for about 15 to 20 minutes.

While the broccoli is roasting, make the bagna cauda: To a mortar and pestle (or a small bowl), add the garlic cloves, lemon zest and lemon juice. Using the mortar, grind the garlic cloves until they turn into a paste. If you don’t have a mortar and pestle, you can mince the garlic cloves and add them to a small bowl, along with the lemon zest and lemon juice. Mix in the anchovy paste or anchovy fillets (if you’re using fillets, you’ll have to mash them with the mortar or a fork). Pour in the olive oil, along with the crushed red pepper flakes and parmesan. Give it a taste and adjust any of the flavors according to your liking. I didn’t need any salt; this will just depend on the anchovy paste or fillets.

Add the mixture to a small saute pan, set over medium-low heat, and cook until just warm, about 1 to 2 minutes. This will cook the bite out of the garlic and lemon just a bit. Pour the bagna cauda over the broccolini and toss until evenly coated. Top with the crispy shallots.