Place beets in an 8-inch square baking pan. Add 1/2 inch water. Cover with foil and bake in a 375° regular or convection oven until tender when pierced, 45 to 60 minutes. When cool enough to handle, peel beets and cut into wedges; place in a bowl.
Meanwhile, in a fine wire strainer under cold running water, rinse capers well, then pat dry with paper towels. Pour olive oil into a narrow 1- to 1 1/2-quart pan over medium-high heat. When oil shimmers, add capers and fry, gently shaking pan occasionally, until buds unfurl slightly and edges turn pale brown and crisp, 30 seconds to 1 minute. Pour into a fine strainer set over a 1-cup glass measure; transfer capers to paper towels to drain.
Measure 3 tablespoons of the reserved oil into a small bowl and stir in the vinegar and salt to taste. Pour over beet wedges and mix gently to coat.
Rinse fennel; chop and reserve a few of the feathery green leaves. Trim off and discard stalks, root end, and any discolored bits from head. Cut head in half lengthwise, then cut halves crosswise into thin, crescent-shaped slices. Reserve 1/4 cup sliced fennel; add remainder to beets and mix to coat.
Mix reserved fennel slices with remaining tablespoon oil. Arrange over beet mixture and sprinkle salad with fried capers and reserved chopped fennel leaves (see notes).