1. In a 5- to 6-quart pan over high heat, bring potatoes and 2 1/2 quarts water to a boil. Cover pan, reduce heat, and simmer until potatoes are tender when pierced, 25 to 30 minutes. With a slotted spoon, lift potatoes out and let them stand.
2. Return water to boiling and add cauliflower; cover, reduce heat, and simmer just until tender when pierced, 5 to 6 minutes. With slotted spoon, lift out florets and let stand. Return water to boiling and add carrots; cover, reduce heat, and simmer until tender when pierced, 8 to 10 minutes. With slotted spoon, lift out carrots and let stand. Return water to boiling and add green beans; simmer, uncovered, until tender-crisp to bite, 3 to 4 minutes. Drain green beans, immerse in ice water until cool, about 3 minutes, then drain again.
3. Meanwhile, in a 3- to 4-quart pan, bring beets and 1 1/2 quarts water to a boil. Cover, reduce heat, and simmer until beets are tender when pierced, 35 to 40 minutes. Drain and let cool.
4. Peel beets and potatoes; cut into 3/8-inch cubes. In a small bowl, mix beets with 2 tablespoons vinegar. Put potatoes in a large bowl. Cut cauliflower florets into 1/2-inch-wide pieces. Cut carrots in half lengthwise, then crosswise into 1/4-inch slices. Cut green beans into 1/2-inch pieces. Add cauliflower, carrots, and green beans to potatoes.
5. In another small bowl, combine mayonnaise, 6 tablespoons vinegar, cornichons, capers, 1/4 cup dill (or 2 tablespoons dried dill), Dijon mustard, dry mustard, and 1/2 teaspoon salt. Add 1 1/2 cups of the dressing and remaining 1/4 cup vinegar to the potato-vegetable salad; mix gently.
6. Overlap gravlax slices end to end to create an equal-size circle on each plate. Mound potato-vegetable salad equally in the center of the circles, surround with beets, and arrange bread alongside. Sprinkle salads with remaining 1/4 cup dill (or 2 tablespoons dried dill). Accompany with remaining dressing and salt and pepper to add to taste.