Preheat oven to 375°. Trim beets, put them on a large sheet of foil, drizzle with 1 tsp. olive oil, and wrap with foil. Roast beets until tender when pierced with a fork, 30 to 60 minutes. Unwrap and let cool.
Meanwhile, in a medium bowl, whisk together remaining 4 tbsp. olive oil, the vinegar, garlic, salt, and mustard. Set dressing aside.
Trim feathery greens and stalks from fennel (save for soup if you like) and slice bulb as thinly as possible. Put in bowl with dressing. Finely chop green onions and add to bowl. Toss to coat with dressing. With a slotted spoon, lift fennel–green onion mixture out of bowl (leaving excess dressing behind) and divide evenly among 6 plates, arranging the salad on one-quarter of each plate.
Peel, seed, and chop cucumber. Put in bowl with dressing and sprinkle with pepper. Toss to coat with dressing. Lift cucumber out of bowl (leaving excess dressing behind) and divide evenly among plates, arranging next to the fennel salad.
Cut kernels from corn and put in bowl with dressing. Chop 4 basil leaves and add to bowl. Toss to coat with dressing. Lift corn and basil out of bowl (leaving excess dressing behind) and divide evenly among plates, arranging next to the cucumber.
When beets are cool enough to handle, slip peels off, cut beets into bite-size pieces, and put in bowl with dressing. Toss with the rest of the dressing and divide evenly among plates, arranging in the remaining quarter of each plate. Cut remaining basil leaves into thin ribbons and sprinkle over salad. Serve with coarse-grain salt, for sprinkling to taste.
Note: Nutritional analysis is per serving.