Prepare fennel: Cut the fennel tops and bottoms. Reserve some of the fronds and set them aside. Cut the fennel bulbs in half, lengthwise. Quarter each half, making sure you leave some of the core attached so that the fennel pieces don't fall apart too much during cooking. I had 5 slices from each half.
Prepare apples: Halve the apple, core it, and cut it into slices as well, I had about 6 slices from each half.
Saute: Heat the butter and oil in a large pan. Add the fennel and cook, stirring occasionally, for about 4 minutes or until slightly golden.
Add the apple slices and continue cooking, stirring occasionally, for another 3 minutes until golden.
Sprinkle the sugar and some sea salt on top. Stir gently for 1 minute until the sugar caramelizes slightly.
Cook: Add the vegetable broth, cover the pan and cook on medium-low heat for further 3 minutes.
Add the lemon juice, remove the lid and cook for 1 more minute until all the liquid reduces to a glaze. Adjust the taste with a little salt and freshly ground black pepper.
Sprinkle with the reserved fennel fronds and serve immediately as suggested above.