Preheat the oven to 350 degrees F and butter an 8-inch square nonreactive metal baking pan.
In the bowl of an electric mixer, beat the butter until smooth. Add 3/4 cup of the sugar and beat until fluffy. Beat 1 egg into the butter and sugar mixture until well incorporated, followed by half of the flour. Repeat with the remaining egg and flour. Pour the batter into the prepared baking pan and bake the cake until a tester inserted in its center comes out clean, about 25 minutes.
Transfer the pan to a cooling rack over a sheet pan and allow the cake to cool slightly. While the cake is cooling, make the syrup by combining the remaining 1/2 cup of sugar with the coffee-flavored liqueur and the reduced coffee in a medium nonreactive bowl. Whisk together until the sugar is mostly dissolved. Using a wooden skewer or toothpick, poke holes all over the cake. Pour the syrup over the cake in several additions, allowing each addition to be absorbed before adding another.
Serve the cake warm or allow the cake to cool completely, and cover with plastic wrap to keep moist. Cake may be made 1 day in advance.
To serve, slice the cake into 8 equal portions. Combine the creme fraiche and the whipped cream, and gently fold together to combine. Place a slice of the cake on a dessert plate and top with a dollop of the creme fraiche mixture. Garnish with a few chocolate covered espresso beans, the mint and dust with confectioners' sugar. Serve alongside the Guinness Ice Cream Float.
Combine the half-and-half, scraped vanilla bean and seeds and salt in a nonreactive saucepan over medium heat. Bring the half-and-half just to a boil then promptly remove the saucepan from the heat.
Beat the sugar and egg yolks in a bowl until thickened and pale yellow. Add the half-and-half mixture, about 1/4 cup at a time, to the beaten eggs and sugar, whisking in between each addition, until completely incorporated. Pour the mixture back into the saucepan and place over medium-low heat. Cook, stirring, for 5 to 6 minutes, or until the mixture becomes thick enough to coat the back of a spoon. Remove from the heat, strain through a fine mesh sieve and cool completely.
Pour the filling into the ice cream machine and process according to the manufacturer's directions. When the ice cream has frozen, just before removing it from the machine, slowly drizzle the caramel sauce into the ice cream as it continues to churn. Allow to churn only until caramel is blended into the ice cream. Remove the ice cream from the machine and place in a freezer-safe container and freeze until firm, about 8 hours.
To make the floats, assemble 8 (1/2-cup) parfait glasses. Fill each glass with 1/4 cup of Guinness. Place 2 (1-ounce) scoops of the ice cream in each of the glasses, and top each glass with enough Guinness to cover ice cream. Place a straw in each of the glasses and serve alongside the Espresso Syrup Cake.
Combine the sugar, water, and lemon juice in a medium heavy saucepan and cook over medium-high heat, stirring, until the sugar dissolves. Let boil without stirring until the mixture becomes a deep amber color, 2 to 3 minutes, watching closely so it doesn't burn. Add the cream (be careful; it will bubble up), whisk to combine, and remove from the heat.
Add 2 tablespoons of the milk, and then add up to 2 more tablespoons, until the desired consistency is reached. Let cool until just warm before serving. (The sauce will thicken as it cools.)