Cut off the stems of the artichokes so they sit without falling over, and trim the pointy leaves with kitchen scissors, if desired. Soak the artichokes in a bowl full of cold water for 3 - 5 minutes and rinse them thoroughly.
Stand the artichokes in a microwave-safe shallow bowl or dish with sides and pour 1 cup of water and the juice of half the lemon over them. Place a plate on top of the bowl and cook them in the microwave for 15 - 20 minutes on high, rotating the dish halfway through, until the artichokes are tender. (An inner leaf will pull easily from the stem.) Alternatively, steam them on the stovetop in enough water to cover them about 1/3 of the way up for about 30 minutes.
While the artichokes are cooling, divide the butter between 2 small bowls and melt it, covered, in the microwave. Into each bowl, stir in 1 Tbsp. olive oil, 1 tsp. grated Parmesan cheese, 1/8 tsp. garlic powder, 1/8 tsp. salt, and a squeeze of fresh lemon juice.
Enjoy dipping the leaves while the sauce is still warm.