Bring a pot of salted water to a boil over medium-high heat. Add green beans and cook for just 2 minutes. Transfer to a bowl of ice water using a slotted spoon.

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Transfer potato cubes to the same pot; simmer over medium heat until tender, but not soft, 5 to 8 minutes. Remove potatoes with a slotted spoon and drain on a paper towel-lined plate.Slide out paper towel and drizzle warm potatoes with about 1 tablespoon olive oil.

Drain green beans. Cut into 1/2-inch to 3/4-inch pieces.

Place the potatoes in a 1-quart canning jar; even out the layer. Add in this order a layer of chopped green beans, tuna, tomatoes, olives, eggs, and anchovy fillet, evening out layers as you go. Top with chopped parsley. Leave about 1/2 inch space at the top. Close with lid and refrigerate while you make the dressing.

Place mustard and anchovies in a mixing bowl. Add vinegar, lemon juice, shallots, tarragon, and avocado, olive oil, black pepper, and cayenne pepper. Blend with an immersion blender until thoroughly emulsified, 1 or 2 minutes. If dressing seems too thick, stir in 1 or 2 teaspoons water or lemon juice.

Pour about 1/4 cup dressing into salad jar, or more to taste. Toss with a fork. Or you can shake the jar and transfer dressed salad to a bowl.