What is it called on the menu? What’s the name of the restaurant(sometimes, things are famous enough you can find them on the internet). You might also try asking the management if the’ll tell you the recipe. I’ve had that happen.
It’s called “Chocolate Lava Cake” - one recipe here but you can find them all over the internet.
You need small oven-proof baking cups, which as a college student you probably don’t have. You can buy Pyrex ones at Amazon for about $10 and they’re a useful item to have around (for exmple, if you have to beat just one egg, they’re the right size).
Note to the OP - this British recipe uses metric weight measures (ex: 50g), as opposed to the volumetric measures that are common in American recipes. (ex: 1/2 cup). If you lack a kitchen scale at home (not that they are expensive) and are an inexperienced baker as you say, I would not even attempt this recipe. In addition to the measurement problems, you may have a hard time finding certain ingredients listed therein.
Rosie’s Bakery in Cambridge makes a Chocolate Orgasm; it’s been part of their menu since at least 1976. It may not be exactly what you are describing, but it’s pretty darn good. You can find the recipe here, or search Amazon for her recipe book if you are looking for more ideas.
I once walked into Rosie’s Bakery with my wife. After we left, I told her that by the expressions on the faces of the female customers women must have a second clitoris in their taste buds.
Our local Italian restaurant serves a version of this (supposedly the secret recipe of the Mama) called ‘Falling Down Cake’, and it comes drizzled with a smidge of raspberry drizzle. I’m also given to understand that the generic term for this is ‘flourless chocolate torte’ (may be helpful in internet searches).
I’ve never tried to make it myself, but need to give it a whirl.