Preheat oven to 375°F (190°C).

Toss asparagus with 1 teaspoon olive oil, rosemary, salt, and pepper.

Spread asparagus on a baking sheet and roast for 10 minutes.

Whisk half and half, eggs, and Parmesan cheese together in a pie plate; season with salt and pepper.

Line 4 cooked asparagus tips on one slice of bread.

Dip the bread in the egg mixture, ensuring the asparagus side is coated.

Heat a saute pan over medium-low heat.

Melt 2 teaspoons of butter in the pan.

Place the bread, asparagus side down, into the pan.

Sauté for about 4 minutes per side, pressing down for even browning.

Flip when golden brown and sauté for another 4 minutes.

Sprinkle the asparagus side with Parmesan cheese while cooking on the second side.

Keep warm in a 200°F (93°C) oven.

Repeat for the remaining slices.

Place all tomato jam ingredients (halved grape tomatoes, 1 tablespoon olive oil, 2 tablespoons sugar, juice of 1 lemon) in a sauce pot over high heat.

Bring to a boil, then reduce the heat to medium-low.

Simmer for 30 minutes, until most of the liquid has evaporated.

Serve the Pain Perdu with tomato jam.