1. Preheat oven to 450°.
2. Blanch beans in a large pot of boiling salted water until just tender, 3 to 4 minutes. Drain, plunge into a bowl of ice and water, then drain again and pat dry on a kitchen towel.
3. Toss potatoes and squash separately with 2 tbsp. oil each, some sea salt, and about two-thirds of thyme sprigs. Roast separately in a single layer on rimmed baking sheets until just tender, 7 to 10 minutes for zucchini and 20 to 25 for potatoes.
4. Make aioli: In a bowl, whisk egg with a pinch of fine sea salt and the vinegar until thick. Whisk in remaining 1 cup oil gradually, drop by drop for the first 1/4 cup or so (until the mixture is emulsified) and then in a thin stream until aioli is nice and thick; you may not need all the oil. Alternatively, whirl egg, salt, and vinegar in a blender, then drizzle in oil.
5. Sprinkle garlic with 1/2 tsp. fine sea salt. Mince with a chef's knife, then use side of blade to crush garlic into a paste. Stir garlic paste into mayonnaise. Taste and add more salt or vinegar if you like.
6. Arrange cooked and raw vegetables on a large platter or long board, top with remaining thyme sprigs, and serve with aioli.
Note: Nutritional analysis is per first-course serving with 1 tbsp. aioli.