In a saucepan, combine butter, oil, sliced onion, and a large pinch of salt. Cook over low heat, covered, turning the onion occasionally, until very soft and it has released its liquid. Uncover, raise heat to medium, and cook until the liquid bubbles away and the onion turns tawny gold.
Add the sliced garlic and cook, stirring once or twice, until pale gold. Add the fine-diced potatoes and turn them to coat, then add enough broth to cover everything by 2 inches (5 cm), or use water plus a bouillon cube. Cover and simmer about 30 minutes.
Add the peas. If using fresh peas, cook another 10 minutes or more until done, replenishing liquid if it drops. If using frozen peas, cook about 4 to 5 minutes until the raw taste is gone. Taste and adjust salt, add a few grindings of pepper, and serve with grated Parmesan on the side.
