Preheat the oven to 450°F and preheat an outdoor grill to medium-high heat.
Half-fill a medium bowl with water and add the lemon juice. Peel and core the pear half, cut it into 1/4 inch thick slices, and add them to the water to prevent browning.
Melt the 1 tablespoon of butter in a small skillet over medium-high heat. Add the diced fennel and cook, stirring occasionally, until it is partly tender and beginning to brown, 2 to 3 minutes. Add 1/2 cup of the diced apple with a pinch of salt and continue cooking until the apple is lightly browned and the apple and fennel are tender, 2 to 3 minutes longer. If they are browning too quickly, reduce the heat to medium. Set the skillet aside to cool for 15 to 20 minutes.
Put the softened butter in a food processor and add the fennel/apple mixture. Pulse until the fennel and apple are well incorporated into the butter and the texture is nearly smooth. (If the butter melts a bit, that’s okay; it will firm up in the refrigerator.) Transfer the butter to a bowl and stir in the chopped fennel fronds and 2 tablespoons of the diced apple. Refrigerate the butter until ready to serve. It should be soft enough to scoop when serving; if it becomes hard, let sit at room temperature for a few minutes.
Drain the pear slices, pat them dry and put them in a small bowl. Drizzle the balsamic reduction over and toss to coat, then arrange the slices in an even layer, slightly overlapping, in a small baking dish. Roast the slices in the oven until heated through and just tender, about 5 minutes. Set aside, covered to keep warm.
Rub the salmon pieces with the olive oil and season with salt and pepper. Grill the salmon until just a touch of translucence remains at the center, about 8 to 10 minutes, depending on the thickness.
To serve, arrange the pear slices on the bottom of individual plates and set the salmon pieces on top. Top the salmon with a spoonful of the apple-fennel butter and a drizzle of balsamic reduction. Top with the remaining finely diced apple and fennel fronds and serve.