The only trick I know of is to use a different variety of apple-- some will turn to complete mush, while others stay firm. Unfortunately, unlike potatoes, they don't tend to be marked at the super market as to which variety they are.
Now, I'm not familiar with Bramley, but from what I've found, it's compared to Granny Smith, which tends to hold up well in pies.
I'd recommend blind baking your crust, if you aren't already, so the filling isn't in there as long, and possibly going with a thicker slice of apple when you're cooking, in hopes they won't break down as quickly.
update : in checking CookWise, there's a comment in the apple pie recipe that "briefly sautéing and poaching apples softens them slighly before the sugar, which prevents cells falling apart and preserves texture, is added"
While not directly addressing the question of apple sauce, this Food Lab article by Kenji Alt on apple pie has some excellent information on differences of apple varieties. He surveys a number of common (at least in the US) varieties.
He notes that apples which brown more quickly tend to be both less tart (as acid, which underlies a tart flavor) inhibits browning; they also tend to cook down less quickly as the acid helps maintain the pectin structure.
Many apple sauce recipes try to find a balance among several apples, with both tart and sweeter varieties, and some that break down quickly for the body of the sauce, and some that retain more texture.
For example, Cook's Illustrated recommends Jonagold, Jonathan, Pink Lady, and Macoun (possible pay wall).
In an article at Oregon Live, interviewing applesauce expert Peggy Acott, writes:
"Even two sweet apples taste different," [Peggy Acott] says. "You can also pair
a sweet and a tart apple, a mild and a spicy one, or spicy, sweet and
mild ones." She suggests using two or three varieties. "One year I
used four and that was overkill."
She believes in tasting the apples before you choose them. The same
variety of apple will taste different year to year, she says. Some of
her favorite past pairings include Ginger Goldens, a creamy yellow
sweet-tart apple, and at least one other variety. Maybe Cox's Orange
Pippins, an old English variety known for being tart and crisp, or a
Rubinette (a cross of Golden Delicious and Cox's Orange Pippin).
An article at the Your pick of apples: Which varieties work best for baking, sauces and other recipes at the Missoulian indicates:
The spicy, supple McIntosh will melt like ice cream when baked, but
creates a smooth, flavorful applesauce. The soft, tangy Jonathan and
the sweet, crisp Empire will also deliver a flavorful puree. The Cox’s
Orange Pippin [...] is a wonderful juicy heirloom for sauce.
As you can see, in the end, you will want to get to know your own locally available apple varieties, and what you like. There is a tremendous amount of variation.
Apple sauce is very simple and quick to make, especially in the microwave. If you are a huge applesauce fan, you might want to make a sample from one apple of each variety and take notes on its flavor and texture.
Best Answer
Walnuts are most likely your culprit. Black walnuts have more of this tendency than common cooking walnuts, but I have had it occur with both. I especially have seen in with port chops stuffed with walnuts and applesauce getting a very purple stain to them. I have seen multiple theories on why purple, but with walnut breads a common suggestion seems to be that iron reacts with the walnut to cause the purple, and some flours are higher in iron than others.
And idea, but just a suggestion, is toasting the walnuts first might reduce this tendency. In fact, here is an article from Cook's Illustrated https://www.cooksillustrated.com/how_tos/6672-preventing-purple-walnut-bread. They indicate that they believe it to be Gallic acid in the walnuts doing it, and that blanching for 1 minute before using them will reduce the effect.
ETA: Looking up Gallic Acid was interesting. It was used as a key ingredient in "Iron Gall Ink" in the 12th through 19th centuries as a standard writing and drawing ink, giving some idea of its ability to color.