Add the eggs, maple syrup, and cornstarch to a saucepan.
Whisk vigorously until the mixture is completely smooth and lump-free.
Gradually pour in the room-temperature milk while whisking continuously.
Place the saucepan over medium-high heat.
Cook while whisking constantly until the mixture begins to thicken and lightly simmer.
Continue whisking until the custard becomes thick, smooth, and creamy.
Lower the heat and add the room-temperature butter, whisking until fully melted and incorporated.
Add the pumpkin puree and vanilla extract, whisking again until smooth and uniform.
If the mixture looks slightly lumpy, remove from heat and use an electric whisk for 1–2 minutes to smooth it out.
Pour the custard into a silicone cake mold or greased baking pan, smoothing the top.
Bake at 365°F (185°C) for about 40 minutes, or until the surface is golden brown.
Allow the cake to cool slightly, then refrigerate for at least 3 hours to set.
Unmold, slice, and enjoy chilled.