Arrange a rack in the middle of the oven and heat to 350°F.
Butter the bottom of a 9-inch springform pan and line with parchment paper; set aside.
Peel, core, and quarter the pears.
Thinly slice each quarter lengthwise without cutting all the way through to the core side, leaving the quarter hinged together.
Finely grate the zest of the lemon into the bowl of a stand mixer fitted with the paddle attachment and set aside; save the lemon half.
Cream the butter and sugar with the lemon zest, occasionally scraping down the sides; add the almond extract and 1 of the eggs.
Beat until combined before adding the second egg.
Beat until combined and then add the third egg, beating until combined.
In a separate bowl, whisk together the flour, baking powder, and salt.
With the mixer on low speed, beat in the flour mixture until just combined, then beat in the juice from the reserved lemon half.
Scrape down the sides with a spatula and give the batter one last mix by hand.
Scrape the batter into the prepared pan and smooth the top.
Gently press the pear quarters into the batter, core-side down, leaving only a bit of space between each quarter.
If you have any pear left over, break off slices and insert them into any open space available.
Sprinkle the top evenly with the demerara sugar.
Bake, rotating the pan halfway through baking, until the cake is golden and a tester inserted into the cake (not pear) comes out clean, 35 to 37 minutes.
Place the pan on a rack to cool for 5 minutes before running a knife around the edge and removing the springform ring. Let cool to room temperature.
Serve with a scoop of vanilla ice cream, if desired.